So after going to the Institute, I saw synth Shaun. After adpoting him and bringing him to Sanctuary, I started wondering if he could age. So, from a RP perspective, in a few in-game decades when my character is 60, would synth Shaun stay the same age, or is he somehow programmed to age?

I'm pretty sure he'll be the same age.

It's like creating a child vampire but worse since synths don't mature mentally.

He never ages. If you walk about the Institute and listen to one of the scientists speak to her family - can't remember who, maybe it was Allie Fillmore - she talks about how cruel and sad it was for synth Shaun. Dr Li, in a holotape, also goes apopleptic at the idea of making a child synth.

Having said that, there's no reason why, in 10 years or so, you couldn't order your scientists to use Father's DNA and clone a 20-year old advlt synth, and transfer those memories over to the new synth. Unless you destroyed the Institute, in which case yes, the poor boy is stuck unless Dr Amari transfers his memories into a new clone, but that new clone won't look like Shaun. Perhaps it would be more merciful to leave him in the Institute as it blows up. There should have been an option to deactivate him rather than leave him there screaming at you - it's not like you don't know his reset code - or does it change?