It's legal for companies to make hiring decisions based on the results of personality tests. It's legal for them to make other personal decisions, such as promoting, firing, giving raises etc. It's legal to make these kinds of decisions based on personality tests that have poor scientific evidence in their efficiency. It's legal to ignore the tests and not make decisions based on them. Largely companies use this as a cover-your-ass tactic. The employer is buying ammunition for future legal fights over employment.

There's really only one set of personality traits (The Big 5) that has a strong scientific backing. Of those only being high in conscientiousness is a signal for being a good employee. Employers screen for that all the time via proxy, such as having a college degree. They don't need to test for it directly. There are however many not-big-5 popular personality tests that are sold to companies. I don't want to call out any specific one, but you should be highly skeptical of their validity.

At best these tests are management by covering your ass. At worst these tests are actively filtering against populations that it is illegal to filter for. For example, the IQ test was once given as part of the immigration process for the United States. Considering that the tests were administered in English they were mostly used as legal justification to turn away non-English speakers.