sys.system_internals_partition_columns view on Azure SQL
Not sure exactly what you want to grab out of the referenced system view. Much of the data you care about is likely still exposed to you, but through the Database related DMVs which have more or less been around in some fashion since SQL Server 2005 with additions in versions since.

You might have some luck with querying a few of these DMVs. You can explore with SELECT * FROM


sys.dm_db_partition_stats
sys.dm_db_index_physical_stats (NULL, NULL, NULL, NULL, NULL) (specifying the parameters required for your usage)
sys.dm_db_index_usage_stats


etc.

There are quite a few database level and connection information level Dynamic Management Views exposed to you in Azure SQL DB.