If an object has both a static ability that causes a player to make a choice for a characteristic-defining ability before the game begins and that characteristic-defining ability printed on it in the same paragraph, those abilities are linked. The second ability refers only to the choice made as a result of the first ability and continues to refer to that choice as the object changes zones during the game.
June 18, 2021 · Modern Horizons 2
If a permanent spell has an ability printed on it that allows cards to be exiled while paying a cost to cast it and the permanent that spell becomes has an ability that refers to “cards exiled with [this object],” those abilities are linked. The second ability refers only to cards exiled to pay the cost of the spell that became that permanent.
April 22, 2021 · Strixhaven
rule 607.2p not present in this release
February 5, 2021 · Kaldheim
Any of the above rules can apply to a planeswalker and an emblem with a pair of linked abilities even though the planeswalker and emblem are different objects.