> They then asked me to provide my address to confirm my identity...I wasn't keen on it.
This means one of the primary avenues of verification (possibly the only avenue for some shops) is unavailable. In the scope of GDPR, it's important to remember that they aren't allowed to retain any information you provide for this purpose for any reason other than keeping a record of the request.
> I mentioned that my full name is globally unique, but they refused.
I would have absolutely no way to validate this, because I don't have a comprehensive listing of all 7 billion-odd people in the world. Even if I did, and it was, it's still only a single factor - I doubt you'd want me to release your data to anyone else based only on them knowing your name and that it's unique.
> My name is...easily searchable, linked to my personal domain, and my personal email address on that domain
This can't be relied on, obviously, because there's no identity verification on (most) domain registrations. For all I know, the email address that I have attached to your profile isn't even yours (because we have no preexisting relationship, this has never been proven.)
> I tried to ask them to share some masked data that I can confirm in full...They refused.
I don't think this is actually allowed under GDPR, but assume it is. Let's say you do this twice with two different data controllers - they each provide you with a masked address, but they've masked different parts (because there's no standard).
If you were a malicious actor, you'd now have the subject's complete address and could use that to gain access to the rest of their data. It opens up a significant attack vector.
> How on earth am I supposed to give all my address history to a company I never heard of, and who shared my data without my consent...
Assuming this was someone unknown and not Acxiom, this is a valid point and unfortunately I don't think there's a great answer. In this case, it is Acxiom and you could've quite easily discovered that they're a major corporation and not a random data harvesting shop.
> I think I was very reasonable, and they weren't.
At the end of the day, you're going to have to give them something to prove who you are. If you won't even provide your old addresses, then absent a government-issued ID (which I assume you also would be reticent to provide on the same grounds) I don't know how else I would even attempt to conduct verification.