To dig into whether Japanese clause-final particle ね and Portuguese term né might be related, we have to lay out where these terms individually came from, and figure out if, historically, there is any evidence of borrowing.
Portuguese né
This word basically means "isn't it?" It is used in a fashion similar to some uses of Japanese ね, added at the end of a sentence to seek confirmation or agreement from the listener.
Various sources online explain clearly that Portuguese né is a contraction of não é, literally não ("not", deriving from Latin non) + é ("is", third-person singular, deriving from Latin est). Compare French n'est-ce pas (literally "isn't it").
What I cannot find, however, is when this contraction first arises. Despite the plethora of sites explaining the derivation of né, none that I've yet read gives any dates for first appearance.
Japanese ね
This word has a lot of meanings attested throughout history, as we can see from the Kokugo Daijiten (KDJ) entry here at Kotobank (in Japanese). Some of these are probably not related to the clause-final particle, such as the ～ね imperative conjugation of completion auxiliary ～ぬ, or the ～ね realis conjugation of negation auxiliary ～ぬ (from older ～ず). So let's just focus on the clause-final particle (technically, the 終助詞【しゅうじょし】). On the KDJ page, this is the section under the headword ね［終助・間助］.
The supplementary notes for this portion of the entry (the [補説] section towards the bottom) explains that most of these senses appear from the late Edo Period, roughly 1600–1868. This is late enough that a Portuguese derivation is not ruled out, considering first contact in 1543 and active trade up until around 1614.
However, one of the senses for ね, a kind of exhortative use like "shouldn't you [do what came before]", is illustrated with quotes from the Man'yōshū, a collection of poetry written from around 600 up through 759 — well before any Portuguese ↔ Japanese interaction. The supplementary notes for ね also mention that this has overlap with clause-final particle な, and the KDJ entry for な includes usage examples from the early to mid-700s.
Looking at the senses for clause-final ね and な, it does appear to me that the modern use of both of these particles, as a kind of "tag question" seeking confirmation or agreement from the listener, has been a natural outgrowth of the oldest exhortative senses.
The history
- The Portuguese first reached Japan in about 1543, and were mostly kicked out in 1614.
- The clause-final particle ね is already in evidence in the 700s.
⇒ Conclusion: The Japanese particle ね and the Portuguese term né are only accidentally similar.
It's actually surprisingly easy to find words in any two languages that kinda sorta sound similar and kinda sorta have similar meanings. If you're interested in word origins and comparative linguistics, I highly recommend you read "How likely are chance resemblances between languages?" The author even lays out a well-backed-up mathematical model for determining the likelihood of such non-cognate similarities.
