Do bartenders try to do this? Yes.
Do they advertise it as such? Yes.
Is it truly original - as in nobody, anywhere, anytime, has ever made that drink before? Maybe.
I used to run a bar, and my bartenders would experiment with these things for various holidays and special events. Customers want to feel special - they know you're not making the alcoholic equivalent of the Mona Lisa, but having a special drink at your local bar is fun and appealing to them.
Once you get beyond the standard cocktails, bartenders (and bars) try to distinguish themselves with variations and twists (pun intended) to stand out in the marketplace.
Combine 1 part each of cinnamon-flavored (some rum) and mint-flavored (some vodka) with 2 parts of (some cream liqueur), shake well with ice, serve it in a martini glass that's been rimmed with (some flavored sugar), stick a candy-cane in it, and dust with (some spice). Call it an "Old Saint Nick" or a "Naughty Rudolph".
Is the Old Saint Nick in XYZ Bar in Birmingham, Alabama, USA the same as the Old Saint Nick in ABC Pub in Birmingham, England? Probably not. Does anyone care? No.