I believe Excel and PBI desktop use an in memory engine taken from SQLServer called xVelocity, as such there is no backend database. The raw data is held in a .xlsx (excel) or .pbix file and loaded into memory.
The system uses very powerful compression algorithms which means large quantities of data can be analysed very quickly and gets around the old excel size issues. There is a 250mb size limit in PBI but because of the compression used 100 million + rows of data can be stored and handled without major issue.
Power Pivot Data Model = Power BI Data Model = SQL Server Analysis Services Tabular mode.
Power BI Desktop and Power Pivot run a private, slightly feature-reduced version of SSAS Tabular behind the application process. The features you're missing vary, but the largest by far is role-based security.
Power BI Web Service uses the same engine - I cannot speak to its specific implementation (what sort of multi-tenancy is used).