"We know that man."
Translation:Znamy tamtego mężczyznę.
"znać" takes Accusative. "mężczyzna" may look feminine (it ends with -a), it itself undergoes declension as if it was feminine, but the fact is that this is a masculine word (after all it means "man", right?)
So the noun phrase "that man" is "tamten mężczyzna". In Accusative -> "tamtego mężczyznę".
Close. 'masculine personal' matters when we're talking about plural.
In singular (only in Accusative), it matters whether a masculine noun is animate or inanimate. "mężczyzna" is animate, so it's "Znamy tamtego mężczyznę". But a dog is also masculine animate, so it will similarly be "Znamy tamtego psa".
In plural, they'd be treated differently. For example in Accusative plural, we'd have "Znamy tamtych mężczyzn" (masculine personal plural) but "Znamy tamte psy" (not masculine-personal plural).