Puerto Rico is not a country

Technically correct — however, while P.R. is not an independent nation, it is also not part of the union. As an unincorporated territory, defining P.R.'s identity is tricky, and highly contextual. American territories are under American sovereignty and, consequently, may be treated as part of the U.S. proper in some ways but not others (i.e., territories belong to, but are not considered to be a part of, the U.S.). As such, places under the U.S. territory umbrella tend to retain their individual "national" identities in the context of international travel regardless of their political status (e.g., similarly to how England/Scotland/Wales are generally considered "countries" despite being parts of the U.K.).