- det = that
- dette = this
It's just how the language works. Similar to English, both det and dette are used but they obviously have different purposes because they are different words. Don't worry, it's OK to ask questions about things you find confusing.
This is my fiancée is a normal sentence to say when you're introducing a person you're engaged with to your parents. Unless you're engaged to a chair, of course. But, that is very unlikely.
In conclusion, this can be used both for living creatures and inanimate objects. Same is true in Norwegian.
As Istor said! For me (native speaker), this sentence sounds more like "It's a man" than "That's a man", because "det" in the sense of "that" would receive more stress (but that's to expect from a digital voice). A lot of sentences that are ambiguous in writing are disambiguated by stress and intonation.