MAME is an emulator running games in fully emulated machines, Proton/Wine is a compatibility layer that runs games natively by the interfaces they expect.

The closest situation would be with comparing Wine running via CPU translation under ARM or RISC-V with how RPCS3 recompiles PowerPC games to native x86 code, but even then the comparison wouldn't be accurate as RPCS3 still does full system emulation whereas Proton/Wine integrates with the underlying OS.

Proton/Wine is closer to Java/JVM than MAME (or any other emulator) and that is when running on a different CPU than x86/x86_64 as on the latter it is just a PE loader with a bunch of DLLs reimplementing various APIs.

It's actually really different. WINE literally stands for: Wine Is Not an Emulator (yes, the acronym is contained within the acronym itself)

Never heard of MAME before :-)