This is sort of apocryphal - and also anecdotal because I have my own personal experience living in an annexed Boston neighborhood to draw on - but in a lot of the towns/neighborhoods that have been annexed by Boston, people still use the neighborhood name[1] as the city name because you are more likely to get your package when you indicate which “Washington St,” “Boylston St,” etc. you actually live at.
According to one commenter on the subject:
It doesn't matter, as long as the zip code is correct
[0]:
https://www.city-data.com/forum/boston/601106-mailing-addres...[1]: https://www.city-data.com/forum/boston/601106-mailing-addres...