Is the usage of "feel like" here correct?
"Now I have lived in New York for seven years, and New York feels like home to me."
You're absolutely right to think that it is correct.
1. feel like = you want to do/have something because you would enjoy it
It's such beautiful weather. I feel like going for a walk.
2. feel like = have a feeling about something/somebody
...and New York feels like home to me.'