The phrase "significant other" is deliberately intended to be very generic, including husband, wife, boyfriend, girlfriend, straight, gay, or non-romantic and non-sexual partner. Because it is a generic phrase it becomes funny when applied to specific cases.
If a man says, to someone who knows he is married, that he will have to ask his "significant other", then this is just a long-winded way of saying "wife". This is a form of comic periphrasis, the use of a long phrase where a short one would do. It is also funny because it is inappropriately generic.
In the Life of Brian a beatitude is misheard as "Blessed are the cheese-makers", and someone says it doesn't only refer to cheese-makers but to all manufacturers of dairy products. Here the interpretation is more generic, but still unexpectedly specific, and that produces the humour.
Another example is where someone was accused of assaulting a police officer with a citrus fruit and a take-away meal. If it is necessary to mention the weapons at all, why not say an orange and a fish supper?. A statement such as "My least favourite mammal is my Uncle Hector" also uses this type of humour. Another example would be to introduce your brother or sister as your "sibling". Sibling is an excellent generic word where appropriate, but is humorously inappropriate to refer to a specific individual.
Significant Other works fine as a generic term, though. Another instance where it can be used without humour is when, for example, a young man says "This is Lisa, my significant other". In this case he means something more than a casual girlfriend, or at least he wants her to think that is what he means.