Unix rule

So Unix is Linux's little brother in his punk phase?

Does DOS even have drivers for gpus with hdmi out?

No, it DOS-NT™
No, but even modern GPUs can run in BIOS and VESA compatibility modes without drivers, which DOS does support. You just won't be able to use hardware 3D rendering.
- It's not even VESA. DOS just uses the VGA 80x25 character terminal mode that all x86 computers still have to start in for backwards compatibility, where "video memory" is mapped to 0xb800 in the 1MB real mode address space. Software you run can then change the video mode, such as to a VESA mode if supported, or for ultra nostalgia, "screen mode 13" (320x200 256-color mode).
- You could also just use a VGA or DVI to HDMI on a compatible card which supports DOS, you'll have to pump in external sound but that's already the case on Most Dos machines who's soundcard is external, and since DOS and its software library doesn't exactly support modern soundcard standards.
I've often thought it would be funny to add more and more modern features to FreeDOS. Funny but like, also a lot of work for a "joke."
You run it inside DOSBox on your Linux installation, simple!
I think FreeDOS does
Lgr did a video 6 years ago where he threw Ms dos 6.22 onto a modern gaming PC just for the hell of it. It has some issues with booting but he it ran. PC specs were a ryzen 5 1600x with a 980ti. And at some point in the video you can see on the monitor he's using that it was using HDMI.
Well it's open source now so it's only a matter of time

A device without any specifications and five different operating system logos behind it. Forgive me, to say that I'm sceptical would be a gross understatement.

Perhaps a more believable post would include the specifications and a link to the supplier.

It’s a KVM switch? How much specs do you need?
- It's unfortunate that Keyboard Video Mouse and Kernel Virtual Machine share the same three letter acronym and that both are in widespread use in relation to multiple operating systems.
  
  Hence my scepticism.
- HDMI Version, max supported resolution & refresh rate for a start
You should be skeptical that those five operating systems are all that it works with. Do you seek out networking equipment that specifically lists compatability with your operating system too?
- KVM != networking equipment (or at least not desktop KVMs)
- I once had a keyboard with a "Works with Netware!" sticker.
  
  Alas, I can't find any networking gear that has a "works with Linux" sticker so I'm just out of luck on that "Internet" thing all the kids are talking about.
I used to have a KVM that was this style. There is a physical button on top to switch between the connected devices, and it listens for double scroll lock taps to perform the same action with the keyboard.

I'd be more skeptical installing any drivers this ali express device comes with.
- I've never used a kvm switch that requires drivers, the fact that it does would give me pause regardless of where it came from