Does modelsim_ae 15.1 support sparse memories?

The clue is in the warning:

** Warning: test.v(4): (vlog-2154) Cannot mark any array in personal edition as sparse.

So you need a more expensive version of modelsim to use it. PE (Personal/Pauper's edition) is one of the cheapest licences.

You forget that the sparse memories you are talking about are controlled by the tool, and allow large memories in code to be run and use less memory without any input from the user. Allowing you to run a large design without running out of ram (if you paid enough money).

Assosiative arrays are controlled by the user and therefore out of the tools hands.