What purpose does a mugshot serve?

Probably to make sure they have the correct prisoner when transferring them from place to place.

It’s a record of your appearance. And, it is used for “wanted” notices if you are suspected of a crime in the future.

If you escape, they know who to look for.

Inside the prison or jail, they know who’s who so you don’t try to say you’re someone else.

To identify a person suing a photograph that is clear and without obstructions in the picture.
Also used in Passports for that very reason.

ID purposes. If an inmate is in prison for a long time, or has a change in weight /appearance, they will retake new pictures.

They will usually document any tattoos, and/or birthmarks. Tattoo photos will sometimes be evaluated by special gang units, to see if the prisoner has any affiliation with known gangs, and may affect many aspects of the prisoner’s needs while in prison.

And it allows you to look up your daughter’s douchebag boyfriends to see what kind of crap they are into.

I see them more as a badge of dishonour, kinda naming & shaming really.