Does the legend rule apply to planeswalkers with the same name or the same planeswalker type?

As of the September 2023 rules update accompanying Wilds of Eldraine, all planeswalkers received the supertype Legendary, and the old 'planeswalker uniqueness rule' (which cared about planeswalker subtypes) was removed entirely. Now only the standard legend rule (CR 704.5j) governs planeswalkers.

The legend rule states that if a player controls two or more legendary permanents with the same name, that player chooses one and puts the rest into the graveyard as a state-based action. Sharing a planeswalker subtype (e.g., two different cards that are both 'Jace') is no longer relevant for this rule.

This means you can control, for example, Jace, the Mind Sculptor and Jace, Architect of Thought at the same time—they have different names, so no legend rule violation occurs. However, you cannot control two copies of Jace, the Mind Sculptor simultaneously, because they share the same name.

Concrete example: You control Liliana of the Veil. Your opponent's spell gives you control of their Liliana of the Veil. Now you control two permanents with the same name, so state-based actions apply: you choose one to keep and the other goes to the graveyard (CR 704.5j).

Remember, the legend rule is checked as a state-based action, not on the stack, so it happens immediately before any player receives priority after the situation arises (CR 704.3).