Yes, it's unlikely that people are illegally voting in person in large numbers. It is relatively easy to do so, and the risk is relatively low, if you approach it intelligently (e.g. vote as someone who is registered, but highly unlikely to vote -- even if they do vote, you're highly unlikely to be caught anyway). However, there's just no incentive for individuals to do so, because the reward is very low: each individual's vote is really worth very little, and an individual fraudulent voter does not benefit from it enough to counterbalance the risk.

On the other hand, there are other ways for people to steal elections. For example, you can steal mail-in ballots from mailboxes, fill them, and covertly drop them in. It's particularly easy to do in states where all ballots are mail-in by default. The risk-reward calculation is different, because now one organized person can cast dozens, or hundreds of fraudulent votes, instead of just one.

In other states, you don't even need to steal them: you can just knock on the door, ask people for ballots (or buy them, many people will happily sell their right to vote for $20, because it's worthless to them), fill them in, and drop them off completely in the open. Of course, the stealing/buying and filling in the ballots is illegal, but since this happens in private, it's much harder to detect and prosecute. That's why most states disallow dropping off votes for third parties, but some states inexplicably allow it.

There are multiple recent cases, where people were convicted for schemes like that, e.g State of Arizona v. Guillermina Fuentes, Texas v. Monica Mendez, Michigan v. Trenae Rainey, U.S. v. Kim Phuong Taylor, and more. Since these are only the cases where conviction was secured, the true number is much higher.

So... For each election, I have to register anew and the agency in charge has a backoffice is cross-checking this against... something? I guess they would first look if I was voting the last time? What if my birth certificate or whatever is from a different place. Do they assume I'm not risking using a forgery over politics (it's a fair assumption I would say)?