What structure is required for my neighbors adjoining fence?
Fences usually have posts of 2 sizes or grades.

Line post are smaller diameter and not as well braced, as they support only fence.

End posts are larger and better braced for extras like gates.
They are meant for tensioning  chain link fences.

If they've attached to a line post, it will probably not support a gate for long.

I would insist that they support the gate with a properly braced post ( usually involving concrete).