To me, the irony here lies in the fact that Tony pretty much achieves the American dream (or is in position to do so) but then he gives it up because he is not comfortable in that life. We tend to believe that people want to move up and out of this sort of neighborhood -- become successful lawyers (or whatever) and move into the mainstream. Tony gets this but finds out that he didn't want it.
As the eNotes summary of this poem says
The irony of the poem is that Tony has the ability and opportunity to escape the Long Island projects, but as an adult he returns to a similar neighborhood because that is where he feels at home.
In the poem, we see Tony getting his chance to go to law school and become more "Americanized." But when he goes to law school in Boston, he can't deal with the different culture. He finds it too strange. So he just comes back to the barrio and lives above the bodega.