The rule is that druids do not wear metal armour. Justification is not given. The GM came up with one. To me this sounds far more immersive than "the PHB says no."
It's a rule that implies fluff. And yes, it is done to reinforce a theme. This is a good thing. The specific implementation of the rule is less good though, as it is unnecessarily vague.
It's bad fluff masquerading as a rule. If the intent was that Druid never wear half-plate, then why give them full on medium armor proficiency and not just say "Hide armor" as proficiency? Or say they can't use wild shape while wearing metal.
It's not worded like a rule, and it's terrible fluff because it doesn't explain squat. It should have been a sidebar explaining WHY Druids won't wear metal armor. What motivates that decision, or at least ask the player to come up with a reason.
The Barbarian not getting their unarmored defence while wearing armor, or the monk losing most of their features without armor? THOSE are rules! Real RULES that DO something. This so-called rule basically takes away agency from the player for no reason at all.
Heck, armor materials are not a rule element. That so-called rule doesn't have any game term to hang its hook on. It's a rule that literally does nothing because it has nothing to interact with and no consequences (like you would if you tried to wear armor without the proficiency). What if we started at higher level and got some spending money for equipment and I buy a breastplate for my Druid and just SAY it's not metal? Can the DM just say 'no'? There's no rules about materials or how much more it would cost. I can just say I had it made out of chitin.
And it's not really fluff because its just a naked statement "Druids will not wear armor made of metal" without ANY story to it.
It's a stupid piece of text.
I have no problem with the concept of limiting what kind of armor a character can wear for thematic reasons, but I object to the sheer arbitrary nature of this particular instance. It's empty and doesn't spark any sort of interesting discussion. Give me something to work with here.