I'm not sure how you originally set yours up but I wouldn't expect anyone to be signing in with their PS4 / XBox IDs. When you first played Elite on the PS4 (assuming it was your very first platform) then it would have asked you to create a new Frontier account which although linked to your PSN ID, is a completely separate account and exists only on Frontier's servers. So I'd expect you to login with the same e-mail address / password that you specified when you created your Frontier account.
I don't have Elite up on a PS4 at the moment but I'm pretty sure the account screen shows you what e-mail address you have set for the Frontier account that you have linked to your PSN ID. So I'd take that the the Frontier web site, use the "forgot password" option, reset the password and then log into it.