No you don't.
I can go to the corner shop/local garage right now, buy a SIM card for 99p and then buy a top-up voucher in cash to have a completely anonymous phone number.
Albeit is the UK in Europe again? 🙈
edit: where I would be worried if my privacy was on the line is I could also go to the local pawn shop / Cash Converters to ensure that SIM card isn't associated with an IMEI I've previously used and buy in cash a cheapo phone.