I would talk to a pharmacist to get an opinion about using an off-the-shelf wart removal system (localized freezing) for this purpose. Those things are cheap and effective at wart removal, but my question is whether the tag physiology is enough like a wart to enable the same treatment. I suspect that it may not be, because while the wart is definitely not “living tissue”, the skin tag may be. So the process that works for a wart may cause injury on the skin tag.
I think that an experienced pharmacist – though she cannot give “medical advice” – would have enough experience with the product to give fairly good general information. And if a doctor is definitely indicated, then that’s the advice you should take.
Regarding your specific question, a “regular doctor” could certainly perform the procedure, but I would ask to be certain that the physician you choose is board certified (first) and has experience in this kind of cosmetic excision (second). Because you don’t want to have to get follow-up plastic surgery to correct an unsightly scar on your neck left from the initial procedure.