The short answer is yes, they can.
But who am I to write about short answers?
I gave our intern a few criteria to screen out therapists to contact: 1) they must specifically state they work with patients with chronic medical illness and/or chronic pain; 2) they cannot describe their therapy style as "eclectic." I cannot stand the use of that word to describe an approach to psychotherapy, and I think when you're shopping around for a therapist you should pause when you see that in their profile.
Let's just start with a simple definition of the word Eclectic: deriving ideas, style, or taste from a broad and diverse range of sources. There were some ancient philosophers who described themselves this way, picking and choosing from several schools of thought rather than adhering to one. Sounds like a good idea, doesn't it? In many ways, being eclectic is a good idea. To a point. But to define your approach to the science of psychology in this way is a problem.
In my graduate school program, we were required to learn the 4 major theories to psychotherapy, each in its own dedicated semester-long course. These included Family Systems theory, Psychodynamic theory, Cognitive Behavioral theory, and Existential/Humanisitc theory. After these introductory courses, we then chose which one fit our fledgling professional style best. I chose Cognitive Behavioral and went on to complete additional coursework, while those who chose Family Systems went along a different path. Our 3rd year we had to present a case as part of our passing the program, and we had to conceptualize the case from our chosen psychological theory. Eclecticism wasn't an option.
If you ask me today, I say I'm a cognitive-behavioral therapist. Do I sometimes borrow from the other 3 theoretical orientations in my day-to-day practice? Sure. To ignore family systems theory when working with a teenager or existential issues when someone is facing end-of-life decisions would be incredibly short sighted. But I would never describe myself as eclectic as some of my peers do.
Wondering if psychology is unique to the use of this term, I poked around the Google to see if it's used in similar fields such as medicine or physical therapy, or in any profession. I did find that in the late 1800s to early 1900s, "eclectic medicine" was a thing where herbal remedies and other practices borrowed from Native Americans were used. The last eclectic medical school closed in 1939. Other than a practice with eclectic in the name, I couldn't find much in physical therapy; same for veterinary medicine. It does show up in fields like music, architecture, and art.
So why do many of my colleagues describe themselves this way? And how many of them truly have expertise in all 4 major areas of psychological theory? I'd never work with a client who was interested in psychodynamic therapy because I'm not good at it. Sure, I took 1 semester-long course on it but I spent the remaining 6.5 years of training doing therapy from a CBT approach. My supervisors were all CBT therapists. It'd be unethical, in my opinion, to take on someone looking to explore their unconscious processes that derive from their childhood experiences. Technically, I could do it. My training is sufficient, on paper anyway; but I would refer them to someone who was an expert on that treatment.
There's a saying an old boss of mine, a physician, used to use - "Jack of all trades, master of none."
Perhaps that resonated with me. When I looked for a gastroenterologist for myself, I sought one with expertise in Crohn's disease. Most people prefer a physician who is confident in their understanding of, and is up to date on, the disease they're going to the doctor to treat. Can they be eclectic in their approach to treatment in that they're open to trying medications, dietary treatments, and stress management? I'd prefer it. But I expect them to also have a solid foundation from which they operate, based on the latest evidence-based treatment and theory of inflammatory bowel disease.
As a therapist, I think it's only right to be able to say the same thing to people who trust me with the care of their mental well-being. Defining myself as an eclectic therapist doesn't impart that.