IIRC Wasps and Harlequins both used to be the same club - Hampstead? But had... Artistic differences between 2 of the founders, so split.
Quins wanted to keep the HFC initials, and Harlequins was kinda the first "cool" name they found flicking through H in the dictionary.
Wasps decided not to keep the same initials (or were forced to, dunno) and did the same at the back of the dictionary.
IIRC Saracens set up initially as rivals to existing local team called the Crusaders (a common enough name for a team without a home ground to start with - see also Wanderers).
IIRC Tigers is from close association with the local army regiment who had a tiger as mascot.
Falcons and Sharks are pure promotional crap invented at the turn of professionalism (1996). Chiefs the same but later, with the added bonus of racism. Warriors the same again, but I can't remember when they changed from being the Pears (from the city crest). (Yes, yes, I know, 100 years Go Exeter named their first XV "the chiefs" for about 3 years - but that's purely an excuse, and most certainly doesn't lead directly to a racist badge, mascot and affectation).
Gloucester Lions and Bristol Bears are even more opportunistic based on money from US firm Gallagher, and a desire to be noticed in America. For a while a decade or so ago, Bristol were called the Shoguns, due to sponsorship by Mitsubishi.
TBF Gallagher did "demand" that all clubs did this, they just didn't actually write it in the contract.
London Irish/Welsh/Scottish used to be both London based, and exile clubs; but that went with professionalism, remaining part of the heritage.
I have no idea for the Saints - but it's a genuinely old nickname - I have a feeling I used to know, but certainly don't anymore.
Currently Bath are the only team without a tacky moniker/badge, though we'll eventually end up as Romans, or Centurions or something (well, it's that or the Georgians...), We already have Maximus the gladiator/lion/centurion as a mascot.
ETA - Saints were started by a vicar trying to "tame" some of the err... Young men with an excess of youthful vigor...
