You wouldn't recognize any religious significance if it didn't host a "gay pride parade." Said parade is a fig leaf.
There are plenty of Orthodox Jews who are disgusted by "gay pride" parades anywhere, but especially in Jerusalem.
I suggest there is no religious significance to having an embassy in a Vatican headed by Pope Francis.
There isn't. With each passing day we see more and more evidence that Pope Francis isn't even a religious leader, let alone a Catholic one. When was the last time you ever heard him mention the name of God in one of his outlandish public statements about climate change, allowing refugees free entry into any country of the world, etc.?