Identify a replacement shimano rear derailleur

As the picture shows the shifter, the derailleur is the bit at the other end, I presume its the shifter that has broken.

Any 7 speed shifter will work. this is a combined shifter so you will need a combined (with brake lever) shifter or you could choose to install a seperate shifter and brake lever.

You will almost certainly need (and be a lot easier) to get new brake and shifter inner cables to install the new levers. The cable installed though the shifter and brake lever then threads back to the derailleur or brakes. Reusing an old cable, is almost always an exercise in frustration followed by installing a new cable anyway. I would install new outers as well when doing a job like this just to be sure they are not a problem. Levers themselves are incredibly reliable, so its possible that the rear derailleur is part of the problem.

Brakes have two cable pull lengths, make sure the lever you purchase matches the brakes on the bike. In this case as the lever says V Brake so any V Brake compatible lever will work.

We don't do product recommendations, but something like ST-EF500-7R2A would be good. If cost is a consideration have a look to see if you have a local bike coop.