I had Chrissy move out, and I ended up seeing her again. When I met her, she had no clue who I was, even if she said she will never forget me. Is that just what villagers say when they leave as well?
In most situations, villagers do not retain any memory of their time on your island. If you encounter them at your campsite (either randomly or by using an amiibo/card scan) or on a mystery tour island, they will not speak of any previous time on your island. They are a new 'instance' of that villager.
However, there is one exception to this. When a villager moves away from your island, they can move to one of your friend's islands. Whether your friend invites them from a campsite visit, or they automatically move in to a vacant lot on their island, they will remember that they were previous residents on your island. If you visit your friend's island and have a conversation with them, they will typically ask if you remember them from their time on your island. I've had this occur personally, and there are instances of this online (e.g. https://amp.reddit.com/r/AnimalCrossing/comments/gwujv9/villagers_moving_into_friends_island_without/)
You kind of answered this in your own question. you even experienced it. A villager that moves out is effectively deleted from the game (unless they move into a friend's island). Any phrases you gave them, furniture, clothes, etc. are all gone.
Any time you meet the islander again is a "brand new" character. This also applies to bottles washed up on the beach.
Probably more because it would be very hard to program and kinda weird if you considered the lore, any villager on the island is your villager ONLY on your island. So my Chrissy on my island is basically a completely different character that looks the same, acts the same but the memories created on your island are proper and only to your island.
When you meet Chrissy again, its another person altogether. Maybe Tom Nook is cloning the villagers and animal crossing as a whole is just Tom Nook creating clones and making sure they work in society and when they suddenly feel the urge to leave the island its Tom Nook throwing them into the normal society as spies. I'M ONTO YOU TOM NOOK.