...except that my druid isn't very religious. He's a philosophical druid, but not a pious one.
Cool, the class doesn't define an orthodoxy, it defines an orthopraxy. If he
will wear metal armor,
he's not a druid, because, as it is explicitly established by the rulebook text, druids won't wear metal armor. But he can believe whatever he likes.
Unless, again, the idea is that every single druid of every single race in every single world from Forgotten Realms where they worship gods to Dark Sun where the gods do not exist, have the same religion.
Well, whether it's the "same" religion or not (incidentally, is there any real-world religion where everybody agrees who is and who isn't a member?), they all have the exact same binding taboo.
You can complain it doesn't make sense, but, does it
really make any less sense than every druid in the multiverse having the same other class features in common? Or the fact that every cleric has the same basic features in common? I mean, yes, it surely
rankles more, but, make less sense?
This is an extrapolation, and a houserule,
The more accurate term would be "syllogism".
The major premise, given by the text of the
Player's Handbook, is "druids will not wear armor or use shields made of metal".
The minor premise, given by the player, is "This character will wear armor or use shields made of metal."
The conclusion follows, "This character is not a druid."
The DM isn't enforcing anything or inflicting a penalty; it's just the inexorable result of cold Aristotelian logic. When the character became someone who
would wear metal armor, he ceased to meet the definition of a druid, so he isn't one.
There are only two valid conclusions we can draw from these premises: either “druids will not wear heavy armor” isn’t a rule, or the rules of D&D 5e violate player agency. I can’t imagine the latter is the intent, especially given that Sage Advice clarifies that it’s just meant as a flavorful thing.
I would suggest that the directly on-point sentence of the Sage Advice Compendium answer is the sentence "If you want to depart from your class’s story, your DM has the final say on how far you can go and still be considered a member of the class."