Screen with built-in NFC
NFC is an inherently narrowband medium, and thus doesn't allow for exact remote location. 

40-50 NFC tags does sound like it could work, but then your figurines would need to be NFC readers – that's a lot of complexity for a figurine.

Conversely, you could put 40-50 NFC readers beneath your screen, but I doubt that's a financially viable option.

I don't know whether NFC is a great approach here. Maybe keep the identification NFC-based, but simply use an e.g. resistive touchscreen and pressure – the moment a new NFC tag becomes visible to single reader that observes the whole table, you look for a pressure point. You know that this must be the new figurine, entering the game. The rest is just tracking the positions that you already know.

Honestly, this isn't a small project – localization of multiple objects on centimeter scale is a hard problem that industrial automation still is fighting with. I'd honestly probably go with a camera mounted across the screen, looking at the tops of your figurines, and some optical object recognition.