I can try and answer some of the questions people have.
Why is it OK to feed/breastfeed for comfort but not give snacks/formula milk for comfort? Should we be encouraging this association?
I was reading a book about BF recently and the author pointed out that she does not like the term "feeding" as in being short for "breastfeeding" because breastfeeding is very much more than just nutrition, and I kind of agree. Perhaps it's hard to get your head around if you haven't done it, or didn't continue past the point where it is strongly about nutrition which is more like up to about a year. I think it's more similar as a process to using a dummy, having a comfort blanket/muslin, or sucking thumb/fingers. All rolled in with a cuddle of course as well. The fact they get a little bit of milk while they do it is sort of incidental, although in an evolutionary sense it makes sense that they still get all those benefits - the antibodies etc delivered in little boosts, kind of like a vitamin shot - even past the point where they actually need to "feed" from mum to sustain themselves. Did you know that when children are older, the concentration of antibodies in mums' milk is actually higher? It's thought this is a response to the fact they are taking much less milk, but their immune systems are not yet mature. The milk isn't needed nutritionally, but that doesn't mean it is without benefit. But it's not really feeding, in the sense of filling up their tummy with sustenance.
So I think the main problem here might be our word for it. Because yes, if you take the term "feeding" out of context of breastfeeding and think about having a little stash of sandwiches or a protein shake or chocolate bar and any time your child is tired/unsure/hurts themselves then you start feeding them with these items then that would be a little bit weird, and potentially a problem in terms of creating an association between food and discomfort. But nobody would think it was weird or inappropriate for a child to have a cuddle or be given their comfort toy/blanket or even a dummy up to a certain age, in that situation. Of course we want children to eventually be able to cope with disappointment, anxiety etc without these things but not at 2/3 years old. And if we're talking about breastfed 4/5/6 year olds, they are typically not using breastfeeding in this way most of the time by that age, even if they might still use it as they fall asleep (as many children of this age and even older still use a comfort toy or blanket or suck their thumb to fall asleep).
Possibly this is why some people use the word "nursing" instead? I don't mind the word "breastfeeding" but I do think the association with "feeding" can be an issue.
Why is it advised to stop using bottles and dummies past age 1 but not breastfeeding?
Well first I'd say I've never come across any recommendation to stop using dummies at age 1, although it's possible I just haven't been looking very hard. It seems culturally acceptable to use them until around 3 or 4 at which point people seem to think it's a bit old - they certainly sell them in shops up to age 24-36 months, whereas I've never seen baby bottles marketed at this age range, more sippy/straw/sports type cups and bottles for toddlers. I have seen that toddlers learning to speak should not be allowed to walk around with a dummy, this is to do with the fact it's preventing them from speaking clearly. It's not really the same with breastfeeding because they can't wander around while attached to your boob - they have to sit still.
For bottles, it's partly about the shape of a bottle and the way liquids constantly drip out of it and can harm the teeth if children have constant access to one, particularly when they have fallen asleep. If the bottle is in their mouth then the contents will pool at the back of the lower teeth and there is a particular cavity pattern associated with this. By comparison, breastfeeding doesn't do this as milk is only released when the child is actively sucking and once they have fallen asleep, they stop. But the other problem is the actual content of the bottle itself. It's likely that a child being given only water in a bottle isn't at particular risk for anything at all, but giving them drinks like squash, juice, cow's milk or formula can be harmful as all of these contain sugars. Breastmilk also contains sugars, but it also crucially contains live enzymes which eat/attack the kinds of bacteria which cause tooth decay, and therefore it is not as harmful to teeth as these other drinks which might be given in a bottle past a year of age. However, I have read that it's important to brush teeth before breastfeeding to sleep as apparently the effect of breastmilk + sugar on the teeth from food or drink the child has had during the day can be quite bad, whereas breastmilk on clean teeth is not harmful in itself.
How does breastfeeding them teach them to self-regulate?
It's not actually appropriate to expect a child of age 2/3 years old to be able to completely self-regulate. They are usually able to do this for some experiences, but others they still need more of a co-regulation approach where someone trusted and calm helps them to calm down and process difficult feelings. Breastfeeding is just one tool which can be used to co-regulate. As children get older, they tend to be more open to other ways such as verbally processing something, problem solving, letting off steam in physical ways etc but children under about 3/4 can't really do this reliably yet. Breastfeeding helps because it's not only a source of comfort as discussed earlier, but also because it triggers hormonal feedback which helps children to calm down, something that they are not always able to do for themselves but will learn to do in time. And as others have said, it's not as though it's the only tool in a box. Someone using it literally for every single little thing would be inappropriate, but that's not very likely, and I think people who make this assumption often don't actually have any experience of feeding an older child and are therefore making an assumption rather than an actual observation.