Cashing out BTC from a BTC ATM that, as means of identification, requests your phone number, will indicate to some institutions that the person who own that phone number has to pay taxes on that?
How this BTC ATM works?
I suspect that it has much more to do with KYC or "Know Your Client" a common term used for complying with United States Treasury Departments anti money laundering concerns. Every entity dealing with potentially larger amounts of cash or cash as a business has reporting requirements. For example any transactions over $1,000.00 must be kept track of. In addition these businesses are supposed to show some sort of effort or procedure in place in collecting information on who they are dealing with. The more cash, the more information they want. A simple phone number likely means a relatively low amount of cash. If you go on Coinbase or Poloniex, the higher the amount of money you want to deal in the more information they want from you. I seriously doubt that asking you for a phone number has anything to do with taxation.
It's not really clear whether those transaction limits are meant by day, week, month etc. Nor it's clear if by transaction is just meant cashing out or the whole action of buying and selling BTC?
I have also noted that (ATMs) have a camera pointing to you, altough it's not clear to me whether they are recording at all the time or not?
If a person cash out their BTC on a regular basis from a BTC ATM after what amount of money this regular cash out can be seen as an income and become "prey" of the United States Treasury Department?
To answer to your last sentence "I seriously doubt that asking you for a phone number has anything to do with taxation."
Well, during this kind of transactions the phone number is needed as means of confirmation of the purchased or sale of BTC. However, if for example, every month you cash out US1500 from an BTC ATM, I guess the ATM owner will have to provide (also )your data to the United States Treasury Department, and they will start to investigate about you right from your personal information, i.e the phone number you provided, and maybe they have the image of your face throught the ATM and see if you declared to the USTD your extra income and therefore pay taxes on it?