TL;DR, the chance of both players leading with the same Pokemon in Gen 8 randbats is a 1/381 chance. This counts alternate forms as the same Pokemon, so Raichu vs Alolan Raichu would count as having the same lead. If you're not satisfied with that, I can try my hand at calculating the probability of just the same Pokemon, same form (although I wouldn't look forward to doing it).
Long answer: First, I looked through the file of Gen 8 Randbat sets and did a quick ctrl+f search to find how many Pokemon had moves assigned to them (since, presumably, Pokemon with no moves assigned to them can't be selected in randbats). But before coming here, I asked on Smogon Forums to make sure the RNG wasn't skewed in any way, and turns out, it is, kind of. Showdown is programmed to weigh each Pokemon evenly when choosing for randbats, but what it counts as "each Pokemon" are separate Pokedex numbers. So, for example, Arceus' 18 forms doesn't make it 18x more likely than any other Pokemon; The chance that you get any Arceus form is just as likely as a Pokemon like Dragonite, but the chance that you get Arceus-Dark, specifically, is 18x less likely, to compensate.
The file that I originally looked through has movesets listed for every form available, so I built a simple program to comb through and only count each valid Pokemon once per form, trimming the number from 460 down to 381. Then we can just take this number outright as our answer; we don't need to square it to account for both players because it doesn't matter what the first player's Pokemon is, the second player just needs to roll the same number.
