This is called list bombing (aka email bomb, list linking, email cluster bomb, subscription bombing, etc). A list bomb is a targeted attack in which a victim's email address is signed up for numerous bulk mail systems without the victim's consent, resulting in lots of bulk mail, ~legitimate and spam, filling the victim's mail box. This got attention in 2016 when Brian Krebs blogged about email bombs aimed at .gov addresses. Wired Magazine also wrote about list bombs on journalists at ProPublica.
In the US, it is still legal to send opt-out bulk mail (a mass mailer can send mail to you until you tell them to stop), as opposed to opt-in mail, which requires the recipient to have previously requested the mail content, or confirmed opt-in (COI, aka double opt-in, DOI) mail, which additionally requires recipients to verify their desire to receive more mail after the initial subscription confirmation.
In my experience, most list bombs are actually confirmed opt-in. They're thanking you for signing up to a list and asking for you to click a link or respond so they have explicit permission to continue.
What are the consequences? List bombing is just a nuisance-level denial of service attack, designed to overwhelm you so you can't do any real work. The only way for an attacker to profit is through extortion ("pay me or it'll happen again") or as a distraction: a list bomb could hide an attack in the noise, burying a critical notice, confirmation, warning, report, the closing letter for a potential business deal, etc.
Why you? This attack probably came from somebody harboring ill will against you. If it's related to a breach, that'd probably just be to gain a list of your users (though there are lots of ways to get users' addresses).
Solutions? Sorry, there is no viable solution to this. It's a real pain point.
M³AAWG (a working group of bulk senders, anti-spammers, and large receivers) drafted a email header to mitigate list bomb attacks, proposing a
Form-Sub header, but the draft expired in May 2020. Also note that this would actually require adoption to be useful.
The only remaining solution, aside from abandoning the address in question, is to design a system that is specifically geared to detecting COI and other bulk mail so you can siphon it off in response to a list bomb event (ideally to be automatically triggered by a sufficient volume and then retroactive to cover what was delivered before the trigger). This is extremely difficult and will be error-prone, so it's something you'll only want to deploy during attacks.
Remedies? If you are actively suffering from a list bomb, consider writing temporary aggressive filters that shove all such mail into another folder for later review as proposed above. This will help limit the risks of missing an important email at the cost of a different annoyance.
Proactively, you could have multiple email addresses (even if they deliver to the same place: forwarders, plus subaddressing, aliases, catch-alls, etc). Important accounts (like banks) would use such addresses. If one of them has a problem, you can filter just it. If your personal address has a problem, you can temporarily filter it to another folder. I don't count this as a feasible "solution" but it does work wonderfully for those who can do it.
I do not suggest changing your address. Since list bombs are mostly COI, they tend to go away on their own, so you just have to weather the storm. Of course, if somebody really hates you, they'll keep signing you up and then you should consider burning the address.
If you have to do that (and you have filter access on your mail server), you could bounce all mail to the old address unless they're in your address book—just make sure that you never reply from the old address since group Cc's won't work. Bouncing (SMTP code 550, "no such user") is important because it'll help inform people you've forgotten to allow.
You could even consider a custom rejection message with indirect instructions on how to contact you, like "try me at firstname.lastname@ instead" (show the formula, not your actual first and last name). I'm guessing even literally including your new email address is safe, though I do suggest mangling it at least a bit.