Li-Young Lee likes to name things. He also rarely seems satisfied with just one name, so he'll give the same thing or experience a number of titles. Sometimes we see it through metaphor, like in "This Room and Everything In It." Or sometimes, like in this poem, he's trying to really pin down a title for an experience, and one name just doesn't seem to cut it. (Another poem that does this is "Little Ache.")
We could look at this as his recognition of the complexity of things, so tangled that they require many names. Or we could call it a love of language and its endless possibilities. Heck, we could call it any number of things, just like Lee would.
Oh, and also: everything about this poem is very, very Lee. From the form (repetition, short stanzas) to the content (a discussion of immigrants) to the tone (meditative), "Immigrant Blues" definitely puts us on Lee alert.