Straight up a determined criminal can do everything including stealing your identity, then you have to go through a lengthy process to recover it, it's not just a simple "hey I'm the real DeTrotse, don't believe this other guy" and even if it were, unless you had family or people to help you out while you sort everything out, everything can be compromised, your assets / money frozen.

And it's not just exaggerating, it's simply the real world. So when in doubt never share things that can identify you, no matter how little. License plates, names of your pets, where you work, school, gym, etc.

If a criminal "just" breaks into your home and robs you, they can at that point also steal stuff to identify you and then steal your identity. And even if you want to be positive, sure maybe worst case they open a few credit cards in your name and ruin your credit.

You also don't have to live in fear, just exercise caution and logic in not over sharing personal stuff.