The case
Every weekday afternoon, the cardigan, the sneakers and the song. Listen closely and Mister Rogers sings "a beautiful day in THIS neighborhood, a beautiful day for a neighbor."
The 2019 Tom Hanks film took the popular version as its title, which locked in the memory for a new generation. The original was always about the specific place you were sitting in.
Why our brains do this
"The neighborhood" is the more generic phrase, and generic phrases are easier to store than specific ones.
Sources: Wikipedia — Mister Rogers' Neighborhood · Wikipedia — False memory: the Mandela effect