Hi OP, parent of ADHD kids here, just some thoughts:
In the supermarket when he's melting down there - do you ever give in to this and buy the toy? Do you notice any pattern e.g. is it likely to occur more in certain shops, at particular times of day, when the supermarket trip is before/after another outing, a mealtime, etc?
The neighbours parents generally let their boy play without a time limit. But we know our son has a limit, and gets over tired. But it’s usually us that calls time on playtime so we look like the bad guys.
Are you worried about looking like the bad guys to your son or to the neighbours? I would not worry what the neighbours think, or if you really worry they think you are uptight, explain to them at a separate time adult to adult that you're trying to help your son with some challenging behaviour. It's also normal IME for different children to have different needs. Probably the neighbours don't need to set a time limit on their son's play. But I would try to set your son up for success. At some point before the next playdate, talk to your son about how you've noticed it's often hard for him to stop playing with the neighbour. Let him know that next time they play together, if he can come home nicely when it's time, there will be a small reward for him - some nice food, TV time, an offer for a parent to do a favourite activity with him, whatever it is. Tell him you will give him a 5 minute warning before it's time to come home, but then you expect him to come home nicely when the time is up. If you want to, present it as a "challenge" with a prize - most children can't resist a challenge.
When the neighbour comes to call or your son goes over to call for him, remind him that he can play for X time, or until Y time (it is useful if he can tell time - this might be something to work on. You can also explain how the time looks on a clock.) When it gets to 5 mins before, let him know and remind him of the deal for leaving nicely. Try to be accurate with the 5 min warning - it's not helpful if you say this and then proceed to chat with the neighbour for 15 mins. Then when it comes to time to leave, if he leaves nicely or there is a significant improvement from normal, give the reward and praise him, because it was likely a significant effort that he made and that should be recognised. If he does not, IME he will ask for the reward and you'll have to say "Sorry mate, that's only if you leave calmly. We can try again next time." He will likely kick off and claim it's not fair etc. Just ignore this/keep everyone safe - don't escalate consequences.
Repeat this a few times and if it works 5+ times in a row, start to fade out the reward but keep the praise for longer. If he objects to the prize going away, see if you can come up with a different challenge for him to work on.
DS will say things like he hates us, we’re the worse , I’m not talking to you. It makes us feel so sad.
Try not to take this personally. 6 is still young in terms of emotional intelligence. TBH even teenagers will say things like this - if you can basically "translate" this in your head to mean "I'm really angry with you right now!! I feel like this is incredibly unfair!! I do not like experiencing the consequences of my actions!!" it can help. It's OK to be the bad guy at times. Unfortunately that's part of the gig. But just the same as children this age will say things like "I am the fastest runner in the world!" they are just exaggerating and extrapolating something temporary to a permanent state because it's a way to express that their feeling is strong. It doesn't mean that they genuinely do hate you, it's just part of how they use language at this age.
Children only tell you that they hate you, they don't want you etc when they feel secure in your love. If he didn't feel secure in his love for you, he wouldn't say it.
I know there’s the counting 123 method but that only really works if we’re home. Can’t really be done in supermarkets or out in public.
You can absolutely do this in public. Only my youngest child really responds to time out but you can just pick any random step, chair or quiet place, or take them back to the car (if the car isn't an oven obviously) or outside the place that you are in. If this works at home, I would use it in public.
Can the demanding things be a compulsive disorder or form of ADHD?
Demanding things is VERY common childhood behaviour, particularly if it sometimes works. Children with ADHD tend to do it more and for longer, because it takes them longer to generalise the rule of not being allowed things, because their emotional regulation can be weaker so they get overexcited quickly and can't easily bring that back down, and perhaps because of sensory overload or less awareness of things like hunger/temperature/tiredness/toilet cues so they may be uncomfortable in these ways and not necessarily realise it but it comes out in their behaviour. But certainly I would not immediately jump to ADHD just on the basis of tantrums in supermarkets. It's much more of an all-round thing so you'd be noticing difficulties, intensity, immaturity etc in almost every area of life.
Like with the leaving the playdate example, IME children do best when you can be proactive e.g. setting the expectation out clearly in advance and reminding them of it shortly before they will be placed in the situation which is difficult for them, so try to use this in any situation which is predictably tricky. For the supermarket one that might look like letting him know before you go in if this is a time he can choose something or if this is a time he cannot buy anything. Having his own pocket money might help, and can also start to teach him about budgeting and spending vs saving. Giving him a specific job to involve him in the trip as a PP suggested can also help.
If he does have ADHD then all of the above will help, because for children with ADHD it's really important to be proactive in reminding about expectations, being positive when they do meet expectations, setting them up for success and always being clear about times. But if he doesn't have ADHD then these are all good things to do anyway, and they might get you to a point where you feel he is more aligned with other children his age.