"I have my husband."
Translation:Mám svého muže.
Exactly. You don't say, I have my husband, unless you just collected or chose him in a mall or contest.
Maybe... your house just blew up and you lost everything in it, but, fortunately, your husband was not home when it happened: "At least I have my husband."
All good. My implied question is, what is the customary meaning of the Czech phrase?
There is no customary meaning. It could have those same reasons you just discussed for English.
Sometimes people will add the unnecessary svého to add stress, but I found very few such examples. Just take it as an exercise for these three words. We may find a better sentence for the next tree.