For some reason it protects like a real armor

Full set protects you not from a bit but almost all damage, yes, it breaks fast, but is it really intentional that rubber makes you invincible for a while?

That is intentional.
Since if you were in a rubber suite it also protects you from a bit of damage too.
But it breaks easily.

Fun fact: Minecrafts armor calculation will only absorb ~80% of the damage iirc, anything extra will be still let through.
Though that isn't as visible since the hearts are a Integer, while the actual health is a float.
Meaning you still can die even in diamond armor with perfect enchantments if you don't have regen and a person punches you long enough.
Without breaking the armor.

But yes this is intended.