Im not an expert in genetics, so someone with more knowledge may come along soon.
They may or may not be the same colour, it will depend on what colour genes the dogs have and whether they are dominant or recessive and if there are any mutated genes present. Not random, don't think horses are either.
Firstly, I'm not an expert but I know a bit, I think there are certain combinations which will always result in the same colour combinations of pups - springer spaniels for instance, liver is recessive so if both parents are liver there would be no chance of a black (or black and white pup). I think labs are the same but I'm not fully sure which way - along the lines of 2 black labs (depending on their parentage) may produce black choc or yellow but 2 chocolates will only produce chocolate.
Generally (to the best of my knowledge) it seems black is a dominant gene so 2 black parents could produce a variety of colours in most breeds
I only know the answer for my breed, & the answer for that is no. I'd imagine it might be same/similar for other breeds. If for example two black GSP's breed (& they would black because they both have a black gene which is dominant) but both possess a liver gene (which is recessive), then any pups who get a liver gene from each parent will be liver in colour (as before, any with even a single black gene will be black). This is before you take into account the coat pattern gene which will change things also. In GSP's you can work out more or less the % of what colour/coat pattern you'll get from a breeding, but nature doesn't always follow theory
Colour genetics in dogs can be simple dominant/recessive in some breeds however then you can get dilute in other breeds which is where things become complicated. There are of course some health issues linked to colours (white boxers and deafness) and the lethal Merle gene which means you should never mate Merle to Merle.
You can get dogs colour tested so puppy colours can be predicted