It's difficult to answer without knowledge about the specifics. However, in Azure SQL Database Accelerated Database Recovery (ADR) and Read-Committed Snapshot are both on by default, which is on the case on-prem. This means that the Persistent Version Store might take up space in Azure.
But as I understand, the databases has different sets of data, so maybe you are simply mistaken about what you think should be the same size. Attached is a file with a couple of queries to give you the size of tables in the database. (The queries handle LOB data a little differently.)