When a male speaker says this kind of thing:
C(fiance): Hey! You sure are getting close to a man's fiancee! That man C is addressing his words to B(other guy).
He is referring to himself even though he says "man".
It is common in English to use this sort of thing. Here are some other examples of what looks like a general statement to another person that is actually referring to the speaker. I am not going to recreate an entire dialog, just the quip:
- You certainly don't know how to show a girl's sister a good time! [for my sister]
- They really love trying a lady's patience. [for my patience]
- She can certainly make mincemeat of a man's heart. [for my heart]
- Do you want to destroy a man's dream? [for my dream]
The word man is used in contrast to, say, a man's heart, a man's home, or something else that is identified as belonging to a man. It is used as a substitute for the more direct: my fiancée. The fiancée is seen as belonging to the man. When this structure is used, there is always a possessive like that.
This is not an AmE or BrE thing. It is a way of not saying something directly for any variety of reasons. In writing, the possessive like this is used quite frequently: a man's life, a woman's life, a woman's home, a woman's children.
[Please note: try someone's patience is an idiom that means testing someone's patience.]