When a matching pair of something, usually bath towels, are labeled 'His' and 'Hers' in order to signify a couple living together. Mostly a Discredited Trope these days, and is almost never played straight.
Parody can come either in the labeling of the objects (for example, "Its" instead of "His" or "Hers"), or in the object itself (say, a military family with His and Hers rifles).
Played with on "The Man Show" in the episode where Adam and Jimmy visit Snoop Dogg's crib. While giving them the grand tour, Snoop shows them the bathroom. On a towel rack is one large towel marked "His" and several smaller towels labeled "Hos." This prompts a question from Jimmy, "How many 'hos' should a 'his' have?"
On Batman, both Batman and Robin have their own specially-labelled fireman's poles to descend to the Batcave. Yes, there is a lot of Homoerotic Subtext.
However, it's worth noting that they go down the poles as Bruce and Dick and enter the Batcave as Batman and Robin, so presumably they're individualised to assist the costume change on the way down.
Game Shows: Dozens of game shows have offered "his and hers" prizes, although said prizes were very rarely if ever specifically labeled that way. Said prizes ranged from watches and luggage to cars. Sometimes, it was clear which was intended for the man and which for the woman (such as watches, by the size of the wristband), but just as often it was two of the same thing with the only difference, if any, being color.
Even in the latter case, if two cars were the prize/part of the prize package and they were not an identical make and model, it was often the sporty-type car that was intended for the man and the more sedate "family" model for the woman.