The spare key marks a kind of belonging that a formal invitation cannot quite reproduce.

Being handed a key says something an invitation cannot. An invitation is for an occasion, and it ends when the occasion does. A key assumes you will come when there is no event, that your presence needs no reason, and that the household has decided in advance to be seen unprepared by you. That is a bigger claim about trust than any amount of hosting.

Consider who has that kind of access to your life, and whether the people you say you are close to actually have it. If someone matters to you, look for the ordinary version of the same gesture, the unplanned visit, the dropping-in without a purpose. Access without occasion is what turns fondness into belonging.