Cannot choose Storage option on a table in SSMS for standard edition
Does SSMS change appearance depending on which SQL Server edition it
  is running against?


Yes

We cannot track everything ssms is executing or making decisions on but we can get a general idea.

Tracing the right click on a table on a SQL Server 2014 standard edition instance will give us more of an idea on the lower amount of checks.



In total, 11 rows/statements found while tracing the action: 

With the options shown:



And when doing the same on a SQL Server 2016 standard edition instance (build version: 13.0.5081.1)



In total, 59 rows/statements: 

And options shown:





In short, on my ssms version (14.0.17119.0) and my SQL Server 2016 instance 
(build version: 13.0.5081.1), the storage option was shown.

I would suggest updating your ssms to the latest version, or using T-SQL instead of  the GUI. 

Patching your sql server instance(s) is also recommended as a general best practice. You could opt to go for SP2 &amp; CU8.



Only for reference, the query that ran on both instances is this one:

        declare @HkeyLocal nvarchar(18)
        declare @ServicesRegPath nvarchar(34)
        declare @SqlServiceRegPath sysname
        declare @BrowserServiceRegPath sysname
        declare @MSSqlServerRegPath nvarchar(31)
        declare @InstanceNamesRegPath nvarchar(59)
        declare @InstanceRegPath sysname
        declare @SetupRegPath sysname
        declare @NpRegPath sysname
        declare @TcpRegPath sysname
        declare @RegPathParams sysname
        declare @FilestreamRegPath sysname

        select @HkeyLocal=N'HKEY_LOCAL_MACHINE'

        -- Instance-based paths
        select @MSSqlServerRegPath=N'SOFTWARE\Microsoft\MSSQLServer'
        select @InstanceRegPath=@MSSqlServerRegPath + N'\MSSQLServer'
        select @FilestreamRegPath=@InstanceRegPath + N'\Filestream'
        select @SetupRegPath=@MSSqlServerRegPath + N'\Setup'
        select @RegPathParams=@InstanceRegPath+'\Parameters'

        -- Services
        select @ServicesRegPath=N'SYSTEM\CurrentControlSet\Services'
        select @SqlServiceRegPath=@ServicesRegPath + N'\MSSQLSERVER'
        select @BrowserServiceRegPath=@ServicesRegPath + N'\SQLBrowser'

        -- InstanceId setting
        select @InstanceNamesRegPath=N'SOFTWARE\Microsoft\Microsoft SQL Server\Instance Names\SQL'

        -- Network settings
        select @NpRegPath=@InstanceRegPath + N'\SuperSocketNetLib\Np'
        select @TcpRegPath=@InstanceRegPath + N'\SuperSocketNetLib\Tcp'



        declare @SmoAuditLevel int
        exec master.dbo.xp_instance_regread @HkeyLocal, @InstanceRegPath, N'AuditLevel', @SmoAuditLevel OUTPUT



        declare @NumErrorLogs int
        exec master.dbo.xp_instance_regread @HkeyLocal, @InstanceRegPath, N'NumErrorLogs', @NumErrorLogs OUTPUT



        declare @SmoLoginMode int
        exec master.dbo.xp_instance_regread @HkeyLocal, @InstanceRegPath, N'LoginMode', @SmoLoginMode OUTPUT



        declare @SmoMailProfile nvarchar(512)
        exec master.dbo.xp_instance_regread @HkeyLocal, @InstanceRegPath, N'MailAccountName', @SmoMailProfile OUTPUT



        declare @BackupDirectory nvarchar(512)
        if 1=isnull(cast(SERVERPROPERTY('IsLocalDB') as bit), 0)
        select @BackupDirectory=cast(SERVERPROPERTY('instancedefaultdatapath') as nvarchar(512))
        else
        exec master.dbo.xp_instance_regread @HkeyLocal, @InstanceRegPath, N'BackupDirectory', @BackupDirectory OUTPUT



        declare @SmoPerfMonMode int
        exec master.dbo.xp_instance_regread @HkeyLocal, @InstanceRegPath, N'Performance', @SmoPerfMonMode OUTPUT

        if @SmoPerfMonMode is null
        begin
        set @SmoPerfMonMode = 1000
        end



        declare @InstallSqlDataDir nvarchar(512)
        exec master.dbo.xp_instance_regread @HkeyLocal, @SetupRegPath, N'SQLDataRoot', @InstallSqlDataDir OUTPUT



        declare @MasterPath nvarchar(512)
        declare @LogPath nvarchar(512)
        declare @ErrorLog nvarchar(512)
        declare @ErrorLogPath nvarchar(512)
        declare @Slash varchar = convert(varchar, serverproperty('PathSeparator'))
        select @MasterPath=substring(physical_name, 1, len(physical_name) - charindex(@Slash, reverse(physical_name))) from master.sys.database_files where name=N'master'
        select @LogPath=substring(physical_name, 1, len(physical_name) - charindex(@Slash, reverse(physical_name))) from master.sys.database_files where name=N'mastlog'
        select @ErrorLog=cast(SERVERPROPERTY(N'errorlogfilename') as nvarchar(512))
        select @ErrorLogPath=substring(@ErrorLog, 1, len(@ErrorLog) - charindex(@Slash, reverse(@ErrorLog)))



        declare @SmoRoot nvarchar(512)
        exec master.dbo.xp_instance_regread @HkeyLocal, @SetupRegPath, N'SQLPath', @SmoRoot OUTPUT



        declare @ServiceStartMode int
        EXEC master.sys.xp_instance_regread @HkeyLocal, @SqlServiceRegPath, N'Start', @ServiceStartMode OUTPUT



        declare @ServiceAccount nvarchar(512)
        EXEC master.sys.xp_instance_regread @HkeyLocal, @SqlServiceRegPath, N'ObjectName', @ServiceAccount OUTPUT



        declare @NamedPipesEnabled int
        exec master.dbo.xp_instance_regread @HkeyLocal, @NpRegPath, N'Enabled', @NamedPipesEnabled OUTPUT



        declare @TcpEnabled int
        EXEC master.sys.xp_instance_regread @HkeyLocal, @TcpRegPath, N'Enabled', @TcpEnabled OUTPUT



        declare @InstallSharedDirectory nvarchar(512)
        EXEC master.sys.xp_instance_regread @HkeyLocal, @SetupRegPath, N'SQLPath', @InstallSharedDirectory OUTPUT



        declare @SqlGroup nvarchar(512)
        exec master.dbo.xp_instance_regread @HkeyLocal, @SetupRegPath, N'SQLGroup', @SqlGroup OUTPUT



        declare @FilestreamLevel int
        exec master.dbo.xp_instance_regread @HkeyLocal, @FilestreamRegPath, N'EnableLevel', @FilestreamLevel OUTPUT



        declare @FilestreamShareName nvarchar(512)
        exec master.dbo.xp_instance_regread @HkeyLocal, @FilestreamRegPath, N'ShareName', @FilestreamShareName OUTPUT



        declare @cluster_name nvarchar(128)
        declare @quorum_type tinyint
        declare @quorum_state tinyint
        BEGIN TRY
        SELECT @cluster_name = cluster_name,
        @quorum_type = quorum_type,
        @quorum_state = quorum_state
        FROM sys.dm_hadr_cluster
        END TRY
        BEGIN CATCH
        --Querying this DMV using a contained auth connection throws error 15562 (Module is untrusted)
        --because of lack of trustworthiness by the server. This is expected so we just leave the
        --values as default
        IF(ERROR_NUMBER() NOT IN (297,300, 15562))
        BEGIN
        THROW
        END
        END CATCH


SELECT
@SmoAuditLevel AS [AuditLevel],
ISNULL(@NumErrorLogs, -1) AS [NumberOfLogFiles],
(case when @SmoLoginMode &lt; 3 then @SmoLoginMode else 9 end) AS [LoginMode],
ISNULL(@SmoMailProfile,N'') AS [MailProfile],
@BackupDirectory AS [BackupDirectory],
@SmoPerfMonMode AS [PerfMonMode],
ISNULL(@InstallSqlDataDir,N'') AS [InstallDataDirectory],
CAST(@@SERVICENAME AS sysname) AS [ServiceName],
@ErrorLogPath AS [ErrorLogPath],
@SmoRoot AS [RootDirectory],
CAST(case when 'a' &lt;&gt; 'A' then 1 else 0 end AS bit) AS [IsCaseSensitive],
@@MAX_PRECISION AS [MaxPrecision],
CAST(FULLTEXTSERVICEPROPERTY('IsFullTextInstalled') AS bit) AS [IsFullTextInstalled],
SERVERPROPERTY(N'ProductVersion') AS [VersionString],
CAST(SERVERPROPERTY(N'Edition') AS sysname) AS [Edition],
CAST(SERVERPROPERTY(N'ProductLevel') AS sysname) AS [ProductLevel],
CAST(SERVERPROPERTY('IsSingleUser') AS bit) AS [IsSingleUser],
CAST(SERVERPROPERTY('EngineEdition') AS int) AS [EngineEdition],
convert(sysname, serverproperty(N'collation')) AS [Collation],
CAST(SERVERPROPERTY('IsClustered') AS bit) AS [IsClustered],
CAST(SERVERPROPERTY(N'MachineName') AS sysname) AS [NetName],
@LogPath AS [MasterDBLogPath],
@MasterPath AS [MasterDBPath],
SERVERPROPERTY('instancedefaultdatapath') AS [DefaultFile],
SERVERPROPERTY('instancedefaultlogpath') AS [DefaultLog],
SERVERPROPERTY(N'ResourceVersion') AS [ResourceVersionString],
SERVERPROPERTY(N'ResourceLastUpdateDateTime') AS [ResourceLastUpdateDateTime],
SERVERPROPERTY(N'CollationID') AS [CollationID],
SERVERPROPERTY(N'ComparisonStyle') AS [ComparisonStyle],
SERVERPROPERTY(N'SqlCharSet') AS [SqlCharSet],
SERVERPROPERTY(N'SqlCharSetName') AS [SqlCharSetName],
SERVERPROPERTY(N'SqlSortOrder') AS [SqlSortOrder],
SERVERPROPERTY(N'SqlSortOrderName') AS [SqlSortOrderName],
SERVERPROPERTY(N'ComputerNamePhysicalNetBIOS') AS [ComputerNamePhysicalNetBIOS],
SERVERPROPERTY(N'BuildClrVersion') AS [BuildClrVersionString],
@ServiceStartMode AS [ServiceStartMode],
ISNULL(@ServiceAccount,N'') AS [ServiceAccount],
CAST(@NamedPipesEnabled AS bit) AS [NamedPipesEnabled],
CAST(@TcpEnabled AS bit) AS [TcpEnabled],
ISNULL(@InstallSharedDirectory,N'') AS [InstallSharedDirectory],
ISNULL(suser_sname(sid_binary(ISNULL(@SqlGroup,N''))),N'') AS [SqlDomainGroup],
case when 1=msdb.dbo.fn_syspolicy_is_automation_enabled() and exists (select * from msdb.dbo.syspolicy_system_health_state  where target_query_expression_with_id like 'Server%' ) then 1 else 0 end AS [PolicyHealthState],
@FilestreamLevel AS [FilestreamLevel],
ISNULL(@FilestreamShareName,N'') AS [FilestreamShareName],
-1 AS [TapeLoadWaitTime],
CAST(SERVERPROPERTY(N'IsHadrEnabled') AS bit) AS [IsHadrEnabled],
SERVERPROPERTY(N'HADRManagerStatus') AS [HadrManagerStatus],
ISNULL(@cluster_name, '') AS [ClusterName],
ISNULL(@quorum_type, 4) AS [ClusterQuorumType],
ISNULL(@quorum_state, 3) AS [ClusterQuorumState],
SUSER_SID(@ServiceAccount, 0) AS [ServiceAccountSid],
CAST(SERVERPROPERTY('IsPolyBaseInstalled') AS bit) AS [IsPolyBaseInstalled],
CAST(
        serverproperty(N'Servername')
       AS sysname) AS [Name],
CAST(
        ISNULL(serverproperty(N'instancename'),N'')
       AS sysname) AS [InstanceName],
CAST(0x0001 AS int) AS [Status],
SERVERPROPERTY('PathSeparator') AS [PathSeparator],
0 AS [IsContainedAuthentication],
CAST(null AS int) AS [ServerType]