Muslims headgear for men functions nearly similar as Jewish headgear . Although Muslims wear it a bit less often. Many Muslims wear it in the house or on the way to or from a Mosque, while prayer etc…
Some Muslims wear it always. depending on observance . I would rule out any alterior motive behind him wearing a skullcap and going to the bank.
My advice to you as a muslim is to treat any religious item as if it doesnt exist . Muslims don’t expect you to praise nor to frown upon the religious items they wear. Just be neautral . Sometimes soem muslim women wear full face veil. If you need to check their identity then you can do so if you are a female , and in the privacy of other females only.
best regards to you in your career