I agree this wasn't good (among other things).
If you entered some wrong login info, which happens, yes, you may have given out some other logins that way.
Link to the hack or not isn't necessarily the question (of course it seems to be the goal here, to make it appear as it caused him some troubles and losses. Maybe also play a little the victim, IDK). For a change of email of the registrant and for a transfert out (Auth code), I'm confident it isn't possible without controlling the current email address of the registrant (or you would have to contact the registrar to change the email because you lost the previous one. They would check a little you're you. Especially with the Epik hack being widely known). Transferring out isn't possible at least without first letting expire the wait period for external transfers (and there are notifications by email. There are some ICANN requirements here, if I'm not mistaken).
I do now have a doubt about internal pushes. Maybe they can be done immediately without controlling the email address of the registrant. I don't remember for sure and i don't want to test a transfer now just to get the answer. I'll see next time I do have some to do.
Or for the story to make sense, it would have to be some inside jobs from the 3 registrars. Logically, anybody would ask questions to them about why the domains aren't in the account anymore while you didn't receive any notifications. Or it is time to worry about your email accounts being compromised, which is dangerous (and this shouldn't come from the Epik hack).