Taking the Isaiah passage in it’s immediate literal meaning, it probably depended on what the current king decided - whether or not he wanted to appoint someone to that position. Our “king”, Jesus, is always current.

The keys in Isaiah concerned a national government office/authority, whereas the keys in the NT are about a spiritual, church office.
When Catholics use the passage in regards to Matthew 16:19 it is to show that “keys of the kingdom” was a term and authority that would have been familiar to the Jews - not a new expression or type of authority that would have been novel and puzzling. Also, taking the Isaiah passage as a prophecy of future events, it shows that the Lord’s establishment of such an office was in fulfillment of prophecy.
I’m not sure what you mean when you say that it’s part of our tradition that “the keys in Isaiah belong to the lineage of Jesus”. Do you mean that the OT kings in Judah belonged to the tribe of Judah and Jesus’ lineage was of the tribe of Judah?
Nita