It’s pretty easy to tell after a fairly short time whether someone “clicks” with you or not. I don’t understand your company’s rule that no supervisors can write recommendations-after all, they know their teams best, but that is what it is. You shouldn’t have to ask if your co-workers like you, though. Surely you can tell?
As for me-I have one co-worker-my boss, and while we don’t hang out outside of work, I just got a glowing review at work so I know I"m liked and appreciated. I work at a church with some members who DO consider themselves my friends, and I feel the same. I also know people from other settings such as my volunteer work, who could write a reference for me if needed. I know they like me based on how they act around me. There are cues.