A formal mental health-related diagnosis is not needed by many therapists. In fact, I think a pretty large number of people who seek out therapy don't have a specific mental illness, at all. Many seek therapy for help with marriage or other relationship issues. Some people are in a life rut either in terms of career, or other life issues and decisions. Some people have various difficulties adjusting to new environments. And so many other issues! Even having some prolonged mood slump need not equal a major depressive illness diagnosis.

I suppose sometimes a therapist (and certainly doctor) wants to "solve" a problem. However, it's not always so very easy. Situations are complex and can't always be approached in a predictable manner. Perhaps it's not a puzzle for a therapist to solve, but for them to simply support. Or to somehow lead YOU to find some solution to your own issue or struggle. Sometimes issues and struggles can never be cured, but rather learned to tolerate, minimize, or look at in a different light.

I do have a formal diagnosis of bipolar disorder, but that's not the only barrier in my life to success, in all ways. Many of us have issues that don't and never will fit a DSM-5 diagnosis criteria. Yes, many of us have anxiety, but not Generalized Anxiety Disorder. Many have little touches of obsessive or compulsive tendencies, but not Obsessive Compulsive Disorder. Most have had some, at least minor, trauma in life, but not Post Traumatic Stress Disorder. Etc. And even without meeting "full criteria", we sometimes need support with such issues.