Tim I don't have Twitter or Facebook so I can't tell how it works there. On other sites you can have both a username and a display name. So I assume this can be done there too.
Normally a setting on 1 site is only for that site. In the bunq app there is no disctinction between a public name that is used on bunq.me and Together and arguably those are used for different things and then could have a different username and a different display name.
Now you have a different username for bunq.me and Together (I have!), but they both use the same display name. That's like having a different username on Twitter and Facebook but having a shared display name.