That's an interesting question. In fact, there have been a number of studies examining the correlation between certain gambling habits/skills and the co-existence of specific mental-health traits and symptoms, positive or otherwise. The findings have covered an interesting, if at times disturbing spectrum, and cast some interesting light on a complex area that we're still a very long way away from fully understanding.
However, in addition to the more formal clinical-research projects, there's also plenty of anecdotal evidence that would seem to suggest a link, or affinity, between certain types of gambler and certain mental skills and/or disorders.
For example, over the course of time the current writer has observed a seemingly way-above-average incidence of Obsessive-Compulsive Disorder (OCD) in the card-counting community. And OCD can often also go hand-in-hand with various depressive disorders.
While we must emphasize that we're not remotely suggesting that all professional blackjack players suffer from some kind of obsessive or depressive disorder, or that we are in any way qualified to write with any authority on this subject, the fact that there might be an overlap doesn't seem so surprising. Consider: Successful professional gamblers tend to have an above-average IQ and a general propensity for analyzing things -- a personality trait that can often be accompanied by a less-welcome self-destructive flipside. There's a form of self-discipline that OCD can impose that can be channeled as an asset by a professional gambler. And a key symptom of OCD is, as the name would suggest, some manifestation of repetitive behavior that can often involve obsessive, but ultimately meaningless, counting. That pretty much defines what card counting is, except that at a blackjack table, you can make money from doing it!
Of course, it's all too easy to fall into the trap of spouting cliches or of drawing sweeping generalizations from a few limited high-profile examples. On the other hand, and again speaking entirely anecdotally but from first-hand experience, you don't have to look too far in the professional-poker world to find examples of prominent (i.e., successful, at least sometimes) players who exhibit obvious signs of narcissism, bipolar disorder, substance abuse, and other traits of the mood-roller-coaster variety. It's not that all professional poker players are irresponsible egomaniacal degenerates ... but if we were betting on whether there's an above-average incidence of the above in that community, we'd go all in.
In fact, at least two well-known poker pros have written candidly about their mental health problems. You can listen here to a 2014 interview with Dutch Boyd for CalvinAyre.com, in which the three-time WSOP bracelet winner candidly discusses his struggles with bipolar disorder and substance abuse, detailed in the autobiographical Poker Tilt, while Mike "the Mouth" Matusow was equally graphic in his memoir, Check-Raising the Devil. The title kind of speaks for itself, but in it the notoriously volatile Matusow details, among other things, his struggles with addiction, bipolar disorder, and attention deficit hyperactivity disorder (ADHD), which he would self medicate with Ecstasy and methamphetamine, which in turn eventually led to his arrest for drug trafficking and a six-month prison term.
To conclude the "downer" section of this answer, we mustn't overlook the now well-documented connection, in some patients, between the use of medications prescribed to treat Parkinson's disease, and the almost-spontaneous onset of some form of gambling addiction. As the Spring 2005 edition of the Parkinson's Disease Foundation's official newsletter put it, just as the link between the two was beginning to be taken seriously, "Add impulse-control disorders such as pathological gambling and hypersexuality to the list of possible non-motor problems that can occur in individuals with Parkinson's disease (PD). Recent studies suggest that seriously disabling impulsive behaviors occur in three to five percent of PD patients, affecting individuals at all stages of the disease. The exact prevalence of such disturbances in PD is not entirely clear since the behaviors are often performed in secret, or because the association with PD seems improbable and is therefore never reported at doctor visits. But as it becomes increasingly clear that some anti-parkinsonian drugs play a role in the development of impulse-control disorders, it is critical that clinicians screen for this possibility and that patients and families report any concerns or changes in behavior to the treating physician immediately."
But enough of the doom and gloom. Returning to the original observation in this question yes, there's good news, too! As an interesting article posted at PokerStars.com puts it (and yes, we realize they're biased, but they're simply capitalizing from valid findings), "While the effectiveness of new and trendy brain-training activities is still up for debate, it is widely accepted that engaging in a wide variety of classic games, such as puzzles, poker, and chess can help increase memory, logic and focus." The same piece goes on to state how "Using rational thought in these types of activities has been proven to increase neuroplasticity, which is a fancy way of saying that these games can help the brain stay organized, understand cause and effect, learn and retain new information, see things from alternate view points, and identify patterns. Increased neuroplasticity also helps decrease depression and anxiety. "
There's wisdom in that old adage that if you don't use it, you lose it, and the converse is true. As PokerStars continues, "Anyone who's ever played poker seriously will tell you that it is one of the most mentally challenging card games in existence. While games such as Gin Rummy can be played practically perfectly once a player masters the game, poker strategy is largely subjective and therefore requires creative and intelligent problem solving in order to be successful. Game objectives such as working out what your opponent holds or how to manipulate them into folding a superior hand are those cerebral gems that make poker the ultimate card game for exercising your brain...
"...Additionally, the game of poker often requires on-the-fly mathematics, memorization techniques (especially in Stud games in which some cards are dealt face up), and overall strategic thinking. Playing multiple games simultaneously in online poker games increases one’s ability to multitask, bringing a whole new level to this intellectually challenging game. On top of all these factors that activate the frontal lobe (the part of the brain responsible for attention and rational thinking), poker promotes patience, discipline, and emotional control in an unparalleled way*."
That last part is laying it on pretty thick and, as this very interesting article for Huffington Post points out, the entire concept of "multi-tasking" is highly flawed. (As an aside, check out Point #6, regarding introversion and extroversion and the associated propensity/aversion for taking risk/gambling, which describes how the two distinct personality types are the result of the brains actually being hard-wired differently). Yet still a valid point is made. Hence, it's of no great surprise that "Godfather of Poker" Doyle "Texas Dolly" Brunson played his last World Series of Poker Main Event in 2013 -- when he finished a more-than-respectable 408th out of a field of 6,352 -- at the ripe old age of 79! The following year he quit playing the Main Event, but not on account of his mental faculties. As Brunson tweeted, "To everybody who keeps asking…I'm not playing the Main Event. Hours are just too damn long.#oldpokerplayersneverdietheyjustfadeaway"
*(Ed: While strategies for playing poker may all promote some admirable qualities, such as patience and self-discipline, that does not mean that they successfully instills said traits in players: Not for nothing did the term "on tilt" originate with the game of poker!)
