To answer the title question: no, not really.
Some background is also here: Does the FAA have a best practice for highlighting memory items on a checklist? (different question, so yours is not a duplicate, even though it's also a "no").
To go into the process a bit more for the airlines at least in the U.S.:
Once the plane is sold to the airline, the second certification process takes place. This certification process (FAR Part 121--large commercial transports) is aimed at certifying the carrier to operate the airplane. In order to prepare for this certification, the fleet manager (or the checklist designer) takes the manufacturer's previously approved procedures and modifies them to coincide with the operational concepts and checklist philosophy of the airline.
The FAR Part 121 certification process is conducted by the Principal Operations Inspector (POI) assigned to the carrier. This individual and his staff are the FAA's representatives to the carriers. They are responsible for initial certification, follow-on procedural changes, and regulatory oversight of the carrier. [...]
With the vague and non-specific nature of FAR 121.315, and with no FAA internal guidelines for checklist design criteria, the POI has almost total authority over the approval of the checklist. He or she may approve or dismiss it according to his or her own philosophy of checklist usage.
— Degani, Asaf, and Earl L. Wiener. Human factors of flight-deck checklists: the normal checklist. Vol. 177549. Moffett Field, California: Ames Research Center, 1990. [emphasis added]
Which makes your other question, "Or is it the case, that each manufacturer / supplier / airline is free to produce their own, with their own unique style?" answerable with yes, as long as it gets signed off on.
I tried to check if the regs were updated since 1990, but I could not find any with specifics regarding the design of checklists. (You might also want to look into electronic checklists.)