There is of course a standard laying down the specifications which an altimeter has to be designed and manufactured to in order to be approved. Unfortunately, it is one of those non-free standards which will set you back a significant amount. I'm on a dumbphone ATM so I won't research the reference, but I think it is in the EASA certification standard documents (and thus probably in the relevant FARs as well). Could have been an SAE one, can't quite recall.
The operational checks recommended are something entirely different.