There are several plausible possibilities.
This is a scam and isn't actually from the public transit authorities, in which case paying them hasn't helped you, and has contributed to this being a problem in the future for others. The payment information may have some subtle differences from the correct information and may actually go to the fraudster. The EBE number discrepancy makes this the most likely scenario in my opinion.
There was a technical error. Maybe someone with a name similar to your was really cited, but due to a typo, your name was entered instead and the street address and corrected name were entered via some sort of autocorrect function. Somebody got lucky and avoided the ticket that should have gone to them. Maybe somebody went into the wrong field in the ticket entry system which could also explain the EBE number error.
Someone has stolen your identity (perhaps a refugee) and is going around with a fake ID using your name and address. This doesn't explain the EBE number discrepancy, however.
I've also heard of cases in some big cities where two people have the same name and birthdate and are constantly getting tickets meant for the other person but don't discover this fact for many years. Maybe someone like that recently moved into your town. This also doesn't explain the EBE number error.
The bottom line is that even though it would perhaps be cheaper and less time consuming in the short run to just pay the fine, I would not recommend doing that in this case.
In scenario 1, you really have a moral civic duty to take a little extra effort to identify a fraudster who is preying on lots of people in your community.
In scenario 2, you again, have something of a moral civic duty to help the transit system get this problem fixed, and who knows, it might be a technical problem that is prone to recur and if you don't address it the first time, people will assume that your "confession" of fault by paying the first ticket undermines your credibility if it happens again.
In scenarios 3 and 4, the problem is likely to recur and so you have an interest beyond this transaction in sorting out the situation.
While your bare assertions that this didn't happen might fall on somewhat deaf ears as potentially self-serving, when corroborated by the fact that you have a monthly pass, I think that the likelihood that you would be found to be credible and beat the ticket is great in your case. Germany is less corrupt in bureaucratic matters like this than most countries on Earth. If this is a scenario 1 situation, the legitimate transit agency wouldn't even have a record of the existence of a ticket in the system, so you could be reassured that you would be at no risk if you didn't pay it.
Given the EBE number problem, it is quite possible that even if you did try to pay it and it was simply a human data entry problem or something (or maybe your ex or a high school bully you defied and forgot about is now a transit cop and trying to inflict revenge) that you might not get proper credit for it anyway, so talking to a person at the agency is probably necessary anyway.
So, my advice would be to talk to a human being in their office, ideally by telephone (without using the ticket to determine the proper telephone number as the number on the ticket would be a scam in scenario 1), but in person, if necessary, to get to the bottom of this. If this didn't work, I would write a letter or email with a copy of the ticket enclosed. If this falls on deaf ears, I would even consider making a police report of a suspected fraud, or even enlisting a local newspaper reporter or television news reporter in pursuing this situation because it is odd and would resonate with the general public.
All of this will be a pain and won't necessary make narrow economic sense, but we don't always have a choice about when duty calls to fix a problem or what problem that will end up being.
