But here's the thing, as others roll before you, you have less options to roll, yes? So at the outset, you all have the same chances to roll 99 (or otherwise get the highest roll). Now consider this sequence of events:
Player 2 rolls a 6, there are now 5 options that you roll below them, but 93 that you roll above them, so the odds are in your favor that you will roll above them (1 to 5/99 versus 7 to 99/99):
Low Roll = 5/99 = 5% High Roll = 93/99 = 93%
Player 3 rolls a 90, so now you have 88 options that you roll less than them (1 to 89 less the 6 previously rolled) and only 9 that you will roll above them.
Low Roll = 88/99 = 88% High Roll = 9/99 = 9%
Player 4 rolls a 80, 90 is still the high roll, but your now have 87 chances you will roll less than 90 (89-1(80)-1(6) = 87), and the same 9 that you might roll above it.
Low Roll = 87/99 = 87% High Roll = 9/99 = 9%
Player 5 rolls a 93, so now you have 92-1(90)-1(80)-1(6)=89 chances that you will roll less than them and only 6 chances that you will roll higher.
Low Roll = 89/99 = 89% High Roll = 6/99 = 6%
Player 6 rolls a 72, 93 is still the high value, but your options are now 88 chances to roll lower (92-1(90)-1(80)-1(72)-1(6) = 88) and the same 6 chances you will roll higher.
Low Roll = 88/99 = 88% High Roll = 6/99 = 6%
Player 7 rolls a 95, so now you have only 4 chances to roll higher than that, but 89 chances you will roll lower (94-1(93)-1(90)-1(80)-1(72)-1(6) = 89)
Low Roll = 89/99 = 89% High Roll = 4/99 = 4%
Your initial odds of rolling a 99 are the same as everyone else's from the outset, but your odds of rolling the highest number decrease as more people roll and the pool of options is further reduced.
Of course it all depends on what everyone else rolls, sure. If they roll 1-7 your odds of winning are 100% as /u/_Fwa pointed out before.
Again, we are talking about two different issues here, one is if you will roll any given number, and the odds of doing so are equal for everyone from the outset. But as the other numbers are taken before you, your odds of getting a higher number than they picked decreases.
tl;dr: First step is determining what number you will roll (1-99) second step is whether or not your number is higher than anyone else's number. That second step does have to at least take into account that as other numbers are chosen your odds of choosing a higher number from the remaining numbers does decrease.