I doubt that St. Mungo is a Christian saint. (Proven wrong by Jason Baker.) If not, is there a religion associated with St. Mungo, or is it an affectation to name it in a similar way to Muggle hospitals?
According to the Harry Potter "fact file" on St Mungo's Hospital, it was
Founded by Healer Mungo Bonham in the 1600s
Which echoes the information found on JKR's original website (and which also gives his birth-date and confirms his gender)
Wizard of the Month - March 2005: Mungo Bonham (1560 - 1659): Famous wizard healer. Founded St Mungo's Hospital for Magical Maladies and Injuries
No indication is given why Mr Bonham was made a saint or whether this was simply a nickname for someone who devoted his life and considerable wealth to the wellbeing of his fellow wizards.
As has been pointed out in comments, there is another St Mungo associated with healing (he resurrected a bird) however this individual died nearly a thousand years prior to Mungo Bonham founding his hospital so it's relatively unlikely that they share an in-universe connection other than a similar first name.