My advisor is currently in hospital waiting to have an operation. It is nothing serious but nevertheless he is away from work. I've noticed that some of my lab mates continue messaging him despite him being in this situation. I wonder if this is a cultural thing and whether this is actually normal? I am in Canada but my background is European if that makes sense.
It's appropriate to contact your supervisor to wish them a speedy recovery.
It's not appropriate to ask your supervisor to help you immediately while they are hospitalized or ill.
It may or may not be appropriate to let your supervisor know you need help with something when they are feeling better.
Personally, I prefer for students to let me know about their progress while I am sick. At least it's more pleasant than thinking about my own progress.
Messaging, yes. Expecting replies, no.
If people tend to "take their work home with them", they may well want to keep in touch with what's going on. Your supervisor may be one of those people, which is why his students/colleagues are in the habit of messaging him.
However you do need to recognise that responses when they're out of office will depend on what they're currently doing. If they're sat watching TV, perhaps they're fine responding to messages. If they're on a date, not so much. If they're asleep, of course not. If they're unwell or going through some medical issue, it will completely depend on how they're feeling.
So if you want to send an update on your status, that's probably fine. If you're asking a question and expecting a reply within any kind of timescale, that's a big no-no.