How to Dynamically change the database using TSQL
SSMS WILL NOT, I REPEAT, WILL NOT SWITCH TO THE CONTEXT OF A USE COMMAND YOU RUN IN DYNAMIC SQL.
If the ultimate goal is to execute some other dynamic SQL inside the chosen database, this is easy enough:
DECLARE @db sysname = N'db1';

DECLARE @exec nvarchar(max) = QUOTENAME(@db) + N'.sys.sp_executesql',
        @sql  nvarchar(max) = N'SELECT DB_NAME();';

EXEC @exec @sql;

If you need to pass parameters, no problem:
DECLARE @db sysname = N'db1', @i int = 1;

DECLARE @exec nvarchar(max) = QUOTENAME(@db) + N'.sys.sp_executesql',
        @sql  nvarchar(max) = N'SELECT DB_NAME(), @i;';

EXEC @exec @sql, N'@i int', @i;

If the goal is to execute some static SQL inside the chosen database, maybe you should consider storing that static SQL in a stored procedure in each database, and calling it dynamically like this:
DECLARE @db sysname = N'db1';

DECLARE @exec nvarchar(max) = QUOTENAME(@db) + N'.sys.sp_executesql',
        @sql  nvarchar(max) = N'EXEC dbo.procedurename;';

EXEC @exec @sql;

And hopefully the ultimate goal is not to run all this code in SSMS just so that SSMS is now in the context of @db... Daniel would really like it if I state explicitly that this is not possible, as @Lothar's comment also stated.