It isn't at all uncommon to come across regional or local manners of address and many of these can feel odd or "more meaningful" to those unfamiliar to it. However these are really no different than the use of sir or miss/ma'am/madam which often seem too formal for people. Fairly common examples that I have heard used in English include love (luv), sweetie, honey (or hun), darling, friend or mate, in addition to the mentioned brother/sister (also bro/brah).
How do I communicate with someone who addresses me in a more meaningful way than I'm comfortable with? To be clear, I don't require that they stop calling me "brother", but rather how can I best respond to them in a way that mitigates any disrespect coming from my end, when I don't wish to address them as "brother", in return.
Their goal is not to make you uncomfortable, but rather to offer you at least a base level of respect in their dealings with you, as they likely do to any customer, especially regular ones. Yes, how they treat you may be part of their daily life according to worldview and/or religious beliefs but it really doesn't matter as you would likely get treated similarly even if it weren't. They are running a service business and want you feel welcome.
Simply be polite and respectful in return in a manner that is normal and comfortable to you (if nothing else, the use of please, excuse me, and thank you are a sound foundation).
The looks I get in return from them sometimes seem like disappointment ... or maybe even hurt.
It is more likely that you are projecting your expectations onto their responses. Again, they are running a business so they want to make sure you know you are welcome, respected, and valued. If they sense they are making you uncomfortable, their disappointment would more likely be based in not providing you the service they intended.
Lately, I've been going to a Middle Eastern restaurant a lot for dinner
Being a regular customer changes the dynamic of the service/customer relationship. They are going to recognize you and you will often get at least slightly better service and more attention than a complete stranger. It is not at all uncommon, and quite natural, for a more familiar relationship to develop in these situations.
For instance, they may start to address you by name, anticipate parts or all of your order (would you like XXX again?), or ask you questions based on information learned during previous visits (like your "how is school" example). Again the goal is to make you feel welcome, respected and valued.