I’m going to say you don’t have the entirely full story.
I just don’t believe a ticket inspector said let me scroll your Vinted account. What I do believe is that he started from the POV of you are travelling on a child ticket and not a child AND you’ve gone a stop too far. And she’s panicked and heard not a child - and immediately wanted to prove that she is right. So suggested ways she could prove it, and this was Vinted etc. He should not have gone through it though, he should have said no, that’s not proof of age.
From my experience with my child, this is where panic takes over. Rationally, I asked him why on earth he didn’t just ask me to send a pic of his passport - but in the moment he wasn’t thinking clearly and panicked.
I say this from experience as very similar happened to my child (who was also 15 and not believed). They were very over zealous with him. They’re not allowed to speak to a parent on the phone for GDPR (again I know this as I happened to be really close as was picking him up at the station, so dashed to where they had detained him and asked these questions in person). At the time when I was on the end of the phone I was really panicked and couldn’t understand why they wouldn’t come to the phone - I thought they were being difficult but when I got there they were lovely and explained they could lose their job for it.
However, what I would challenge is - the fine will be an on the spot penalty. It’ll be the same whether you have no ticket/the wrong ticket.
So they already had the wrong ticket (went a station too far). He decided to be a jobsworth and issue a fine (you don’t have a leg to stand on with this as they did have the wrong ticket - where people tell you that inspector X let them away with it etc, that’s because inspector X was not a jobsworth, was a decent human but actually, from a job POV they actually didn’t do their job properly, and this inspector did). Therefore whether she was 15 or not (and the subsequent phone scrolling) is irrelevant.
If the fine was based on age alone, you could/he could argue that this was an attempt to verify age and therefore not fine her - fair enough (ish)? But he was fining her anyway so this is where I think you have grounds to say abuse of power/overstepping.
His counter argument would be the above, so you would in your complaint immediately debunk that theory by pointing out that the fine was being issued regardless.
He should have very calmly said - in future, please carry proof of your age. You could be fined for travelling on a child ticket if you cannot prove you are a child. However, because you are travelling on the wrong ticket, I am issuing you with a fine for that anyway - here you go. That would have saved the drama and panic and confusion.