Klingons mate. Humans mate too. They might label it "making love", "having sex", "shagging", "doing it", "making the beast with two backs" or something else to try to differentiate it from what the rest of the biological world does, but it's still mating.

Sure, but what we're interested in is labels, or words. Outside of a science-fiction context, nobody speaking modern English says mate to refer to people having sex.

ngagh and nga'chuq seems to refer to basically the same sort of act. The question is, do they get more specific?

Or is it primarily used for animals ?

Humans and Klingons are animals. 

But languages usually distinguish between people and non-people, and Klingon basically does this in its capable-of-using-language suffixes and its pronouns. The distinction here may be important in Klingon. It is in English.

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SuStel
http://trimboli.name