If it says "too many accounts" then it is likely Apple's restriction. I don't know that your financial institution knows how it is being used at Apple. They just see a charge is being made at Apple, not the ID of the person doing that. If your financial institution thought there was fraud happening with a card they would just block it and all Apple would see (and report to you) is a a negative response from your bank when they tried to charge a card.
I could see a restriction being placed as a fraud prevention measure. Let's say your card was stolen. You wouldn't want a thief then giving it to a dozen friends so they could have a shopping spree? However, only Apple can tell you for sure. If the won't then it is understandable they may be reticent about sharing their security measures.
Refer to this page for Apple Support features ➔ Contact - Official Apple Support
Select from the presented options until you find a solution for your issue, or see if there is a chat or phone call contact method offered lower on the page (you sometimes have to narrow down the options multiple times before this is shown). If you do not see your issue, keep experimenting with any series of selections until you reach one that offers a chat session or a telephone call and get the representative to redirect you.
This page has telephone numbers for contacting Apple for support and service ➔ Contact Apple for support and service - Apple Support
I suspect that most families use Family Sharing which means that although up to 6 people have access to a card, it is only present on the Organizer's account.