Would you take responsibility for this mistake?

lala

Hi so long story short, i work at a dry cleaners where we also do military alterations. When we get an order we get the ticket with the ticket number and tag the clothes with the ticket number as well. Whoever took the order has the ticket under their name. And sometimes when it's super busy we grab the clothes and the ticket and put them down on this table so we could go finish the other orders. And when it calms down when start working on the orders, regardless who's it is.

Side note: we sew on patches on military jackets and take them off with a blade. If you mess up and rip the jacket, you have to pay to replace it.

Anyways so apparently a jacket under a ticket with my name was tagged with the number of another order. Those jackets cost $200 and they replaced it with another jacket they had around (keep in mind, this customer could be a completely different size) and just did alterations on that random jacket. I have been training in that place for almost a month now and since I'm new and the ticket is under my name, of course i got the blame for it. And if the customer notices it's not his jacket (which is most likely because he could be A different size than the original) i would have to pay $200 to replace it.

However, seeing how sometimes we work on each other's orders to clear them after it gets busy, i do not want to take the responsibility of doing that and i am refusing to pay. Of course I've made mistakes when working there, and i ALWAYS let them know but i think it's unfair i might be the one to pay just because my name is on the ticket. Even tho there's a chance it wasn't me who did it.

Plus, i am training part time so my pay checks are not a lot and i can't exactly afford it.

Am i being childish? Am i overreacting?