There are three issues here:
- ties go to white
- there are an odd number of rolls
- the tiles all start white
At first glance it appears the first and last of these give an advantage to white.
Working backwards from the last tile, it will be black if an odd number of sixes are rolled, and white if an even number are rolled. On any two rolls there are 36 possible combinations. Of these, 1 is two sixes, 10 are a six and a not-six, and 25 are zero sixes. Even wins 26 to 10. There are an odd number of total rolls, and on that roll it's 5-1 there will be an even number (0) of sixes. So overwhelmingly (almost 3 to 1) the last tile will be white.
The second last tile will be black if the total number of 5s and 6s rolled is odd. 16 results involve only 1 through 4 so have zero 5s and 6s. 16 involve just 1 5 or 6. 4 involve both. So even wins here 20 to 16. Closer than the last digit, but still advantage white.
For the "4" tile, you need an odd number of 4/5/6 rolls in total. Still looking at two rolls at a time, of the 36 results 9 will involve zero 4/5/6. 9 will involve only 4/5/6 - an odd number. 9 will involve only 1 and 9 will involve 2. So here's it's 50-50 chance whether that tile ends up white or black.
I am skipping the "2" and "3" tiles for lack of time. My intuition is that they match 6 and 5 respectively, but with while and black switched. So overall the chances of "4" being white or black is 50-50, and whatever advantage white has on "6", black has on "2", and whatever advantage white has on "5", black has on "3". For those last 5 tiles, it's 50-50 who will have more of "their" colour.
Here's where I found myself surprised:
The 1 tile will flip every single time. Roll a 1 - you flip it. Roll a 2 - you flip it. Etc up to 6. It flips 99 times which is odd, therefore it will be black.
The rules are therefore slightly less unfair than they first seemed. In reality you are dealing with 5 tiles (2 through 6.) If three or more are white, white wins. If three or more are black, black wins (because of the guaranteed black first tile.) Since the odds of each of those 5 tiles being white or black is 50-50, the game is (surprisingly) fair with each player having an equal chance to win.
If there were 98 or 100 rolls, this would not be the case.