Is it possible to get a research assistantship between undergraduate and masters degree, while not enrolled as a student?
Sure, it is normal to work for a university lab with only a bachelors or masters. However, none of the people I knew did this as a transition between undergrad and grad, rather they did it to transition out of research.

The job titles range widely though, especially across fields. I have seen job titles of "Research Programmer", "Research Engineer", "Software Engineer", "Systems Researcher", etc. This is in Computer Science so other fields may have something along the lines of "Lab Technician" or other domain specific titles.

If you want to do research, then apply to graduate programs that are relevant to you and attempt to contact potential advisors.