On the second thought, I think the keyword is "become" or "стать" in Russian, which is used similarly and sometimes interchangeably to "быть" but needs the instrumental case, so the стать-behaviour has projected to the "быть"-verb.

This is what your car has become. - Вот, чем стала твоя машина.
This is what it's supposed to become in the future - Вот, чем это станет в будущем.

So both incarnations and excarnations are handled using the instrumental case.
(Твой подарок на мой день рождения был велосипедом means that it was a bicycle but now it is already not, it has excarnated and turned into something else).
Твой подарок мне на день рождения был велосипедом, пока я не дал его брату покататься...