How do they know who I am from my phone number?

It may have been from one of their partners such as B&Q or from google analytics that provide info from website browsing data.

The supermarkets seem to be on a marketing rampage just now.

Last week I answered an ad offering a voucher from Tesco. I don't normally interact with ads but this was Tesco... my local supermarket.

I got so far filling it in before I realised it was just a phishing exercise, Tesco passing on info to a data management company called Aura media.

I know this because even though I left the site for a few days I got a lot of cold-calls on my phone which is very unusual. One chap was trying to sell me Life Insurance, but I had a few questions about where he had obtained my number. He was very open and honest about it and promised not to phone me again.
The calls have tailed off now but I think this maybe a similar kind of phishing for the things you are buying.

When we make on-line interactions our date gets stored and used by these data companies to persuade us to buy more stuff.

My advice would be to stop using scan and shop, unless you don't mind your data being used like this.

There are websites you can put a name & phone number into and you'll get a list of people who fit those details. Maybe the supermarkets have access to a variation of that service, tailored to their needs. If you ever thought Big Brother was a myth, you can forget it now. We're far more exposed than even suspicious people can imagine! (I'm no conspiracy theorist, btw.)

I think that when I initially set up my Asda Scan and Go account, using my phone number, I had to give my name.

This is the price we pay for having services like WhatsApp. My OH spoke to a firm in Kosovo for over an hour today for no charge but they have to make their money somewhere i.e. selling data

Marydoll

I think that when I initially set up my Asda Scan and Go account, using my phone number, I had to give my name.

Exactly

That is strange. As a first time user you would normally have to input your name and address as well. I can only assume that somewhere along the line you've signed up for something or other, maybe with one of their partner organisations, and forgotten all about it. It's easily done and could have been some years ago and for something quite trivial.