Learn Polish: Wydawało mi się, że zwiało państwu namiot. - It seemed to me that it blew your tent away.

Because namiot isn't the subject here—it's the direct object (in the Accusative case). Zwiało is an impersonal neuter verb used for uncontrollable natural events. There is no actual subject doing the blowing; it's an invisible 'it' (the wind) acting upon the tent.

Why does the sentence use the Dative państwu instead of just saying państwa namiot (your tent)?

In Polish, especially with uncontrollable events, it's much more natural to use a Dative experiencer to show who was affected by the action. While zwiało państwa namiot isn't wrong, zwiało państwu namiot (literally 'it blew the tent away to you') is more idiomatic because it emphasizes that this unfortunate event happened to those people.

Both halves of the sentence use neuter past verbs without a clear subject (wydawało, zwiało). Do they follow the exact same grammar rule?

They are very similar—both are impersonal, use a 3rd-person singular neuter verb, and take a Dative experiencer (mi and państwu). However, zwiało specifically represents a physical force of nature. Wydawało się is a standard fixed expression meaning 'it seemed,' used for mental states rather than physical events.

What is the infinitive of zwiało, and why is it perfective when wydawało is imperfective?