What would prevent someone else from requesting the deletion of your account if there was no proof that you are the person whose account it is?

it's not illegal to put account deletion behind a login at all. Its also legal for them to request identification.

However if you request data deletion and they have no valid exception to avoid doing that they must comply, it doesn't matter if they have a mechanism for deletion that you can use, they have to still delete the data even if you don't press the "delete account" button.

you can file a complaint with your countries regulatory departments but if they refuse to press the delete account button for you, there's not much else you can do outside of that.

It has already been said but the company is complying in the sense that it is providing a solution whereby you can delete your account. That said, where you are unable to follow that process, they should offer you the same ability via email. Each company does things slightly differently but I would hazard a guess that an email stating that you find it more reasonable for the action to be carried out via email, they would be likely to comply.

The reasons why companies put these in place is simply to avoid mass requests for deletion and, as stated, to also protect you.

While email spoofing has been mentioned, it is somewhat unlikely anyone would send a request for deletion after spoofing your email, yet, it is not impossible.

You may have to be persistent, could use services that support or even get some pointers from the ICO. Here is a really good link https://ico.org.uk/for-the-public/your-right-to-get-your-data-deleted/

I wish you luck

saying when the account was created (month and year)

This is an absurd requirement, but do you have a ballpark idea, and does it let you continue to guess multiple times? Submit a few dozen password reset requests and if they complain, tell them to verify you via alternate means.

Did you tell them that you don't have access to your account? Because to me, this reads like the default answer ("here's how you do it yourself"), not like they won't do it by email if asked to do so.

My company does the same - if you ask our support how to delete the data, they will send you a link to the customer portal. Only if you tell them that doesn't work for you, they will work out another way.

We built the feature into the customer portal, so we want people to use it if possible. Because that's way less work for our support team.

Password reset requires saying when the account was created (month and year) and "tech support" can't help here either.

Did you try the date of their first email?