I do so love the Oxford Unabridged Dictionary:

I looked up "belong". As has been suggested, it does not imply "ownership" necessarily.

"To be connected with in various relations; to form a part or appendage of; e.g. to be a member of a family, society, or nation, to be an adherent or dependent of, to be a native or inhabitant of a place;..."

An example that the editors give that seems to be from what I guess would be roughly contemporaneous with the song is this Shakespearean use of "belong".

1613 — Hen. VIII, v. iv. 3 Good M. Porter, I belong to th' Larder.

Perhaps this helps.

Sourdough