So your Railcard (the part taken from you by staff to prevent further use - for obvious reasons if it no longer showed an expiry date) would indeed have looked a bit like the one you have shown and should have had a date under 'Expiry Date' - a few questions that may help people give you the best advice
a) are you really sure that the bit of the Railcard you gave up had no sort of date under the 'Expiry Date' section - from when you bought it? (this is relevant but given you no longer have the card or an image of it you can't prove that to the Railway now - ie that it was a mis print, although it would seem likely that it might have once had a date but in the subsequent years this has rubbed off - if you failed to read the date and understand it's expiry then I don't think you have much of a defence on that point...)
b) You say you got it when you were 17 but are now aged 22. Roughly when (just the year) did you buy it and thus when did it expire - 1 year later ?
c) Have you ever had another Railcard since? (I don't understand why they told you when stopped that it was valid in October 2023 if it expired about a year after you bought it when you were 17
d) Which Railway company are you dealing with now - in terms of who has written to you?
e) They are asking for: "£625.90 for 5 rail tickets which add up to £61.85."
so this is 61.85 worth of tickets and a fairly large admin fee to settle the case? (it might help to upload the letter you have had without your personal details showing)
If you had no railcard that was valid since you were 18 this might be to cover a fair few journeys, but they are only asking for 5 tickets worth of journeys.
So the question to me is what is the make up of the balance that gets this to £625? That may be a question you could ask them but as this is an unusual case it will probably help if you weigh up other people's thoughts on how to progress this. there is no doubt you will be asked other questions so people can try to understand what has happened and give you the best advice on the way forward/ what to do.
I'll try to answer you're questions as best as possible to my knowledge, unfortunatley the information they told me to what I've recieved 6 months later is different which is why I am confused.
a. I am certain it had no expiry date from when I bought it, this was something I expressed to him while he recorded the interview. He even commented that usually rubbed off would be the case but he said it looked so clean and I said I have never had to provide it to anyone for years there would be no rubbing that would get rid of all the ink, smudges included which he agreed.
b. I bought it when I was 16 or 17, most likely 17 so I would assume expired in 2019, but on the system when he took me to the ticket office to pull the data from the card it said it was valid from October 2023. This was one big confusion for me and both staff members.
c. No railcard since which is why they were more confused why it was valid.
d. Great Western Railway
e. So this is more confusing as he asked me to show my trainline account history which only had 6 tickets, then in the email I have recieved it now says a total of 16 rail journeys "between 22 September 2019 to 18th July 2023." so this can be the additional costs and I am trying to find this information but I don't understand where it came from as I was told the railcard had no history in person.
I think my biggest issue is I understand the reasoning and I have potentially been guilty due to negligence as a teenager, but the problem I'm having is none of the facts are adding up from what I was told inperson to now being sent a bill. I assume I can request the evidence of the inspectors recording and the card he took?