I find soft play overwhelming, and I am 42 years old! You need to realise this is a baby and he is acting to get his needs met. If he wants cuddles and reassurance at soft play or in the supermarket, the best thing to do is give them. I'd also try to learn from his behaviour about what he can and cannot tolerate. Some children find supermarkets too stimulating, and so they act out when they are there. This is very normal, and the solution if you are lucky enough to have two of you available, is for one of you to do the shopping without the child. He is too little to regulate himself in an overwhelming space.
Similarly: Don't take him to soft play if he doesn't enjoy it, or take him home when he has had enough — my DD often liked to stay for 15-20 mins and then got grouchy because she was overstimulated, so we went and had lunch or a nap. He may suddenly find he is tired, or hungry or bored, or constipated, or scared by some new thing, and he is too small to manage these feelings and recognise how to cope himself, so he seeks reassurance. If you give him the cuddles he wants and slowly transition to moments where he needs to wait, or not be picked up etc he will find it easier to manage — but you have to do it when he is not tired, or overwhelmed or feeling unwell so you are only asking him to cope/regulate emotionally when he is fully able to do that.
He will grow up and become more capable of coping with overwhelming environments with time and work from you. But your job as a parent is to provide a safe space from which he can slowly venture out and then return when he feels he needs safety again, and this will be a process that happens bit by bit until he is an adult and forms new primary attachments. You are expecting too much of a very small child, he will be learning to regulate himself and his feelings over the next years to come. It will not be linear and you need to not worry if he regresses because this is normal as he develops independence incrementally.