Exactly this. When me and OH started living together, I made it crystal clear that we'd be doing the "chores" equally and started that habit on the day we moved in. He was under no illusions that I'd be doing anything more than my fair share. Going back even further really, when we used to go on holiday together, I made him wring out and hang out the washing, make the bed, wipe around the bathroom sink after brushing teeth, shaving, etc. I do the washing up, he does the drying and putting away. We do the garden jointly too - either of us will mow the lawn, pick up leaves, trim branches, etc. Same with washing the cars. Not a single thing is "his job" "my job" - it's BOTH our jobs and whichever one of us will do it when we see something needs doing!
You need to "train" them early on!!!
Set boundaries. Make it clear that if they see something messy, bits on the floor, etc., then get out the vacuum or cloths without having to be told. It becomes ingrained behaviour.
Same with "life admin" like paying bills, getting quotes for insurance, phone/tv/broadband contracts, organising car MOTs and services, etc. Either of us does that kind of thing, as and when needed. Nothing like that is "my" job nor "his" job.
My father was a very "hands on" Dad and did loads around the house, so I just assumed it was normal for men to do similar to the woman. My first serious boyfriend was a total lazy slob - his flat was an unbelievable dirty mess. I soon binned him as I wasn't putting up with any of that! Luckily, my OH saw things my way and that's why we're still together 35 years later!!