Their monument sticks like a fishbone in the city's throat.
Fishbone in throat, eh? That seems pretty uncomfortable! A fishbone in the throat would feel like a sharp knife.
By "city's throat" Lowell might also be thinking of the city's anatomy— in this case, its center.
Using this simile to describe the monument does not seem positive. Monuments are supposed to be good, right? They're typically a way for a city to honor its history, but this description makes it seem like the monument is a kind of a pain in the neck (get it?)—not positive, and not comfortable. Perhaps Lowell is hinting at Boston's conflicted history with civil rights.
Its Colonel is as lean as a compass-needle.
In this simile, comparing the Colonel to the compass needle, we see that the statue is actually lean and potentially sharp-looking, like one of these. So maybe this fishbone thing is also referring to what seems like the sharp, pointed shape of Colonel Shaw's statue.
Using "its" at the beginning of line 32 is perfectly grammatically correct, but it's a little dehumanizing. It makes us think much more about the object of the statue, rather than the people it commemorates. (And it kind of reminds us of Cousin It from the Addams family, too.)