I tested with a stored procedure. I copy the stored procedure with the same name but I added the "sp_" prefix, so I have one copy of the stored procedure with the prefix and the other one without the prefix. To make sure I made the test on equal conditions I run both of them with a cold memory (after scaling up and down the service tier) and after clearing the procedure cache.
The elapsed execution time of the stored procedure with or without the the prefix was consistently 77ms or 78 ms.
Based on my limited test the performance impact of using "sp_" prefix that we see on SQL Server instances, does not apply on Azure SQL Database.