I just checked the one in my vega and it doesn't have a brand either. Looking at RV parts catalog it seems the major brands are Valterra, Phoenix and Catalina. And they show knob-type handles and the handle-type for faucets. Does not show anything for individual handles.
Suggest you take the busted handle to a major plumbing outfit (Ferguson comes to mind) or Lowes/HD and see if they have something you can use. Other recourse is to replace the entire unit,
The sink faucet in our 2016 Vegas corroded badly, and I replaced it with a single handle Moen faucet from our house. It's a standard faucet, and much better constructed than the OEM one. It fit perfectly.
If all else fails, read the directions!!