Think are a lot of different terms / combinations for these things - time cards (was the London Transport term), running cards, running boards and I've heard other terms. Also, some operators didn't / don't issue a physical board (or whatever) per bus which stays on the bus all day, but one per driver (a duty board, duty sheet, duty card depending on where you are) which stays with the driver until the end of their duty. Some have issued both.
Yes, each bus will have a board number / running number somewhere, although it's relatively rare for this to be visible from the roadside. And again the sequence can vary. As
@TheGrandWazoo says, a system like 25A upwards is not unknown, where buses generally stay on the same route all day. With other operators, it's a sequence of numbers, and there's a lot of ways it can work - it may be that 1 is the first to leave the depot and so on, or each group of numbers could be for are a particular route / group of routes, or double deck workings might start from 1, single deck workings from 101, and so on.
On London Transport, each bus has a running number that also identifies the garage (in London, motor buses traditionally have garages, depots were for trams and trolleybuses) and each batch of numbers is a particular route - the bus in
this photo is showing 'BW 152' - BW is Bow Garage, running numbers BW 151+ were used for the 'heritage 15' at that time. Usually, a gap is left between each sequence, so one route might be BW 1 to BW 23, the next route would start at BW 31 or even 41, to allow room for allocation changes without having to re-number everything (and there might be a gap if a route had been withdrawn or moved garages.) The lowest number in each sequence may or may not be the first to leave the garage, traditionally the system worked so that for the bulk of the day, buses would follow each other in running number sequence. An increasing number of London operators don't have running number plates on the side of the bus any more, though - their principal purpose was so that roadside inspectors could see what working each bus was on, and that's mostly done by more high-tech methods remotely now. (Up until a certain point in time, each route would start from 1, so there could be multiple buses showing 'BW 1' on different routes, but that was phased out some years ago. Tech systems wouldn't be able to cope with that now.)