Off the more general description in the OP, I was ready to say “it’s probably insecurity.” People tend to react more aggressively when they feel like they’ve got to defend or hide something, I think…
With the more granular details… it could still be insecurity, or partially insecurity, or not really insecurity. It sounds like they’ve got (as @Cruiser suggested) 20 years’ time to have developed a tense situation around household upkeep. (I know that’s the case in my family, anyway!) What seems like one small issue with a wine glass probably seems to him like yet another piece of some other issue (however justified or unjustified,).... alternatively, it might seem connected to a larger issue to her, but he might insist it’s an isolated thing, and he’s reacting to what he interprets as an unreasonable reaction.
He might think “well, if she wants them hand washed, then she can do it.”
Or “of course, there’s yet another thing I can never do right.”
Or “I’ve worked all day and earned the right to wind down with some wine, and all she cares about is ragging on me about one lousy glass.”
He might have had a different chore arrangement with his previous wife.
He might really think that the dishes aren’t his “job.”
He might not see the point in washing a single dish on its own.
He might think they glasses ought to go in the dishwasher.
He might not mind dishes accumulating in the sink.
He might feel “out of place” doing the dishes and so avoid them.
He might resent being told how to clean, and so resists.
He might think he’d only mess it up or not do it the way she wants.
He might be used to not thinking about chores (especially if he’s used to the idea that women clean the house).
He might not realize just how much of the housework she’s actually doing.
He might think the work is split evenly, and doesn’t realize how his apparent dumping of work onto her would feel to her.
Conversely, he might not realize how often he does this sort of give-her-the-chores behavior, and then resents how “strongly” she “inexplicably” reacts to this “one small thing.”
He might feel generally insecure about how well he does things, and since he’s gotten pushback on this, it might have become a sore spot.
It might be the fact that she doesn’t drink alcohol and he does every night, too—if he’s one of those people who doesn’t feel comfortable being the only one drinking, or if he thinks for whatever reason she is irritated by the drinking itself and not merely the dishes.
There might be baggage from an earlier point in his life.
Has she tried the “when you do x, I feel y?” trick, to try and get him to see it from her perspective, or even better, to open up a conversation about it (and what might be underlying it)? Maybe, “when you leave the glass in the sink, I feel like you’re expecting me to clean it, and while I thought it was fine initially, I’m beginning to feel frustrated/taken for granted/etc.?”