Ok, in my universe he sang "It's a beautiful day in the neighborhood." He didn't sing "it's a beautiful day in this neighborhood." This implies it's just a nice day in his neighborhood. The whole idea is that in this kids show everyone has a beautiful neighborhood. Anyway. Don't agree with me?
Eddie Murphy does a skit lampooning Mr. Rogers Neighborhood called Mr. Robinson's Neighborhood. In that skit Eddie sings at the start "It's a beautiful day in the neighborhood."
Here are some examples
Now, why would Eddie Murphy sing the wrong song not once, but 4 times in 4 different skits over 2 years?
Glad I cracked the code on this one!