I encountered this in a computer club (I mean the place where you play per hour to play computer games on a good PC if you lack a suitable one at home). The accounts there were using phone numbers as usernames, and apparently if one is used, it would have to be verified. However, after refusing I was just given one-time accounts every time (with a random string of digits as the username), I just couldn't save unspent time for another visit so had to pay precisely. Funnily enough, the host herself the first time mentioned one-time phone number rental services for this reason)

Best not to overthink it - The sales clerk is trained to ask for this stuff.

Luckily most times I encounter this I just tell them no I don't have a phone number with them & continue checkout like normal. Sometimes that means not getting a sale price on something but usually I avoid those type of member-specific sales anyway.

And worst case - Just make something up. At Best Buy a sales rep absolutely refused to sell me something from the mobile dept without my info. Which didn't make sense because earlier I had bought something at that same Best Buy with a different rep & that rep took my order without my info no problem (she said she had to enter a phone number but just entered Best Buy's).

Yet this particular sales rep refused to proceed without info, so yeah he got an entire fictional name/address/phone/email on the spot.

A long time ago they wanted a phone number for me to get a hair cut. She looked at me like I was crazy when I said no.

I realized a few years ago that my GF inadvertently solved this issue for me: She likes registering for anything that provides a discount, so I use her phone number.

"Are you a member?"
"Nope, but my GF probably is..", and 90% of the time I am correct.

I asked nicely why do I need to give my phone number and I was told that to register me as a member so I can get the discount.

I declined and said I don’t want to join and would like to just pay.

I've just said "I don't have one" when asked this for awhile. This never seems the phase the cashiers, I'm guessing they know what that really means. Half the time I still get whatever discount, though I've never tried to sign up for a membership saying that.

If it's an online form my phone number is just (local area code)555–5555. I've never had that not take, except for one case where it automatically enabled 2-factor auth and I had to create a new account.

Here it's similar. I've found out that if to the question

do you have our membership?

I respond "No, thank you", they often understand correctly and don't assume I want to set one up

I usually just say "Nah, no time, just ring me up."

If you're visiting from another country, try giving them a number from there (real or fake). What are the odds that their system can cope with international codes?