It's easier than that. They are NY teams because they were founded in and spent a combined total of 75 years in the city. Then, seeking a new stadium to play in, they both ended up in the Jersey swamps 5 miles west at the Meadowlands because it was cheap to build there. The boundary between the city and Jersey is by far the most blurred sate line in the country.
If the Giants hadn't moved to NJ 47 years ago (and the Jets 39 years ago), then yeah, bringing this up in 2023 might still be funny or a real burn to those 2 teams and their fans. They really don't get how this makes fun of them at this point. If you live in the city, you already knew that.