The standard translation of to belong to someone is jemandem gehören — no preposition.
Diese Villa gehört mir, diese Jacht gehört mir, dieser Privatjet gehört mir auch. Der Maserati gehört aber meiner Frau.
Typically, this is used to indicate possession. And possession is typically associated with objects, hence jemandem gehören is usually not used with people — and if you use it for people it may sound offensive.
Das sind unsere Sklaven. Peter gehört mir, Paul gehört meiner Frau.
Zu jemandem gehören is a different phrase. It does not mean to belong to somebody in a possessional sense but rather to belong (close) to somebody in a locational sense.
Er gehört zu mir, wie mein Name an der Tür.
The singer doesn’t want to possess him, as she might want to possess jewellery or money. She wants the person to be close to her and feel at home there.
Similarly for the name: she doesn’t want to claim possession like one would for a brand name. Instead, she wants the name to be seen as connected to her.
The phrase can be interpreted as very similar to:
Er gehört hierher
This example further exemplifies the locational aspect of the expression.