My psychologists never did a psychological test because they assumed they knew me in less than first session - Is that normal/okay?

Yes - it's okay that your psychologist didn't do a psychological test. Not all psychologists will - some can spend the first session or two just getting to know you and building that trust. That doesn't necessarily mean that they think they know you; everyone just approaches things a bit differently.

That's not okay. I'm not sure if that's normal however I found doubt it. You can't know somebody within one session, you have to have follow up visits often enough, this is because you may say something on the next visit that you didn't on the last that could be critical in helping to give you a diagnosis.

Depending on why you were seeing a psychologist, a test may not have yielded appropriate information. Even if you had taken a test, a psychologist would only be able to interpret it properly after getting to know you better.

no psychologists need to take tests to be sure guessing may lead to a incorrect diagnosis so to be safe ask the psychologist to do a test anyway its the best option for your mental health

I am not sure what you mean by "they assumed they knew me in less than first session" as the reason they did not do a psychological test. Therapists use an very wide range of ways to evaluate you well beyond formal writen or standardized tests. Everything from body language, appearance, speech patterns, affect (the emotions you show) can be part of evaluating you. They also need to get your history. All this and more would be concidered 'normal'for a psychologist to use. If they start with a psychological test like the MMPI can take 5-6 hours and give very specific information that might have little use in your therapy would not be normal.