It’s an extra output on top of any transaction. The size depends on the length of the name, but it’s quite small. If you choose a 4 byte name “owen” then it would be 42 vbytes in total to register. That’s 4200 sats fee, or $1.54 at current prices + the cost for the rest of the transaction (it can be any transaction as I said).

In the long run it seems like there will be some need for scaling, but for now a few bucks is pretty reasonable for registering a name. I can't believe these brc20 bros keep relentlessly throwing all their sats at the miners. I assume they will run out of sats at some point and fees will relent.