Eidolon (specifically I'm talking about unchained, but they don't seem to differ in this regard) can have an armor or natural armor bonus:
Armor Bonus: The number noted here is the eidolon’s base total armor bonus. This bonus can be split between an armor bonus and a natural armor bonus, as decided by the summoner. This number is modified by the eidolon’s base form and some options available through its evolution pool. An eidolon cannot wear armor of any kind, as the armor interferes with the summoner’s connection to the eidolon.
If one chooses the armor bonus, does it mean that the eidolon can be summoned with what looks like an armor and is detachable? It would still technically be a part of the eidolon, and it would vanish if it is banished, but meanwhile, could it be used as an armor by another creature?
For an Eidolon, having an armor bonus is an intrinsic part of the creature. It in no way implies that it can be removed.
I completely agree with Dave