Keith Stone, the poor man’s Most Interesting Man in the World.
Only for one particular rich man.
Not really. While the gimmick of each is loosly about a fake psychic solving crimes, that is about as close as it comes. They are two different shows with different tones, appeal, and premise. The Mentalist is a former “psychic” who now uses his skills but mostly tries to debunk the notion of psychics (except when he can use it to shake up a suspect). Psyche is about a guy who pretends to be a psychic to mask his real talent, mostly because when he first got noticed by the police, they didn’t believe his explanation for how he could call in with solutions to crimes just from watching TV and wanted to arrest him as complicit in the crimes. So he made the psychic thing up to cover his own ass, and it stuck. The Mentalist has elements of humor but is mostly a straight up criminal procedural, whereas Psyche is a comedy about crime solving.
Okay, that’s probably overprocessing, but the point is the analogy isn’t particularly accurate. One is not the poor man’s version of the other.
[We now return you to your regularly scheduled program.]
Rugby, the poor man’s football.
Porsche 914, the poor man’s Porsche. 
George H.W. Bush, the poor man’s Joseph Kennedy.
The Partridge Family was the poor man’s The Monkees, which in turn was the poor man’s A Hard Day’s Night. The Archie Show was the poor kid’s Partridge Family.
Brad Pitt, the poor man’s skdo23.
Zombie thread from 2008, the poor man’s zombie thread from 1999.
Diogenes the Cynic, the poor man’s Collounsbury.
I was already to make this Kinkade joke again!
Somebody call 911, the homeless man has a Gandalf! (late, I know)
Epic.
Music preferences, the poor man’s politics.
Politics, the poor man’s [insert dumbest current reality show here].
Geraldo Rivera, the poor man’s Mike Wallace.
For voice acting:
Michael Dorn, the poor man’s James Earl Jones.
Stanislaw Lem, the poor man’s Jorge Luis Borges.
Kirk Cameron, the poor man’s Christian Apologist.
9/11 truthers, the poor man’s JFK conspiracy theorists.
Birthers, the poor man’s truthers.
Poor men, the poor man’s friends.
I mean this in the nicest way possible and as a good hearted ribbing, not trying to be mean or flame anyone. Apologies in advance:
Rand Rover, the poor man’s Bricker.
Ambivaled, the poor man’s handy
Message boards, the poor man’s having some smart friends over for dinner.
Been nearly a year, but thought I’d mention that as my wife was watching Grey’s Anatomy, I caused her to bust a gut with my observation that Kevin McKidd is the poor man’s Daniel Craig.
Adolescent Radioactive Black-Belt Hamsters, the poor man’s Teenage Mutant Ninja Turtles.
Dark Star, the poor, poor, POOR man’s 2001: a Space Odyssey.
Bagpipes is the poor man’s cow. According to Paul Lynde.