Well, yes you can “just walk into the DMV and say hey it mine”. Actually, it’s a little more involved than that but basically, you fill out a special form on which you swear that it is in fact your vehicle, they check the VIN number to be sure it is not stolen and then it gets registered in your name.
Some years ago I bought a 1958 Edsel that was from out of state and did not have a proper title. I was able to get it registered in my name by doing just that.
Doesn’t work like that here in Oregon. You can drive around as long as you want with expired tags, you just have to look out for the police. Here in the city of Portland, you have to pass DEQ. And if the vehicle is registered in P-Town the state will inform you that you need to retest. But I bought the truck down in Woodburn.