By the way, this isn’t a login process involving a web browser or a username and password. Usually, if you’re running a Let’s Encrypt client on a web server, the account is authenticated using an encryption key that the client generates and saves somewhere. As @JuergenAuer suggested, if you still have access to the server, there may be a client application that may contain these credentials.
But also, Let’s Encrypt determines eligibility to issue certificates based on control over DNS records, not based on the Let’s Encrypt account. If the old account can’t prove its control of the site, then it can’t renew. If the new account can prove its control of the site, then it can renew. That’s why you can also just create a new account if necessary.
In general, Let’s Encrypt accounts are not very important for most purposes. They are created automatically by client applications and they don’t have long-lasting (or exclusive) power to reissue certificates. There are some limits on duplicate accounts, but the limits are very high and most people would never be anywhere near reaching them except as a consequence of a misbehaving automated script.