With a dedicated DAC + AMP, you can use the DT 1990 without any problems. With regard to the PS5, however, it depends on how you have connected it (or can connect it at all). I only have a PS4 Pro myself - and therefore cannot say with 100% certainty about the PS5 - but it has HDMI + optical output (the normal PS4, without Pro, only had HDMI output). As far as I know, the audio signal on the PS's goes via the HDMI cable (and to the controller), but on the PS4 Pro you can change the settings so that the audio goes via the optical output.

According to pictures of the PS5 on Google, it seems to have only one HDMI output. The easiest option would probably be to install Playstation Remote Play on your PC, connect it to your PS5 and then stream the PS5 on your PC. Then you can listen to the audio via the DAC + AMP in the DT 1990 as usual. Alternatively, if you want to send the audio directly to the DAC and listen to it in the DT 1990, you would probably need to hook up an audio extractor or something like that to separate the audio signal from the HDMI cable and send it separately (e.g. via optical, coax or RCA) to the DAC.

As Dark Dragon wrote: feasible yes, recommended or "just so" possible rather less.
The headphones are great, but there are certainly more suitable for the PS5 (Corsair HS80 or if you want to spend a lot, the Audeze Mobius). However, the latter is way over the top for games (my opinion).