Why would you do this?

[MoroccoMole]

I've found out my ex has registered his car to my address after a tax reminder came through the post. He bought the car 4 months after he left why would he do this?! Can this affect me at all? Is this some kind of fraud?! AIBU to write to the DVLA

Write to the DVLA. As it's not in your name, I don't believe it will affect you personally OP but it sounds dodgy so let the DVLA know ASAP.

It’s fraud, he probably did it because it’s cheaper at your address.

You need to get it changed. If for any reason he doesn’t pay - ends up with a CCJ etc. it will be registered against your address and during your tenure at the property. While not in your name could still cause you hassle if you try to get credit / a mortgage etc.

He probably hasn't changed his driving licence address.

Maybe he needed finance and put down your address as he wouldn't have got it if he hadn't put your address as that was where he was registered at that point as being kn the electoral register and then forgot to change it later.

The VRD would have come to your house too when he bought it.