From a purely mathematical standpoint; these are not equal concepts.
Assuming fair forms of dice, a d100 is a pure set where each face equals its face value in probability.
So, 55 on a d100 equals 55% chance of hitting 55 or lower (which is the primary point of % methods - "at or lower").
2d10's has two main schools of thought: one holds that 90 on the tens and 0 on the ones = 100, and the other holds that 00 on the tens and 0 on the ones equals 100.
Neither method equals the probability of a 100% fair die (if one has issue with the physical die due to physical flaws, compare to a digital 100 sided die - e.g. a random number generator from 1-100).
2d10's does not equal a 100 sided probability because 2d10's coupled in this manner (that is, not summed) functions under the same rules as dual test coin toss.
If you take a coin, you have a 50% chance of getting heads or tails.
If you make a rule that you need to hit heads on two consecutive tests, then your probability isn't 50% (1/2), it's 1/2*1/2 = 1/4 = 25% chance.
This is akin to how 2d10's as "percentile" work.
If your target number is 55, then you need your tens throw to be 5 or less before you can consider your ones throw valid.
Your one's throw is this considered at 5 or less.
The ones is dependent on the tens die and both need to succeed to grant a success.
Depending on which way you treat the die (90|0=100, or 00|0=100), your probabilities vary.
Take the above example.
In 90|0 = 100:
Your tens die hitting 50 = 60% chance
This is because your values at or less than 50 are, 50, 40, 30, 20, 10, 00.
Your 5 on the ones has a 50% chance, however.
This is because 90|0 treats the ones as 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 (10).
Just as we multiply on the coin toss we multiply the d10 results because the second test (ones) is dependent on the first test (tens).
So that becomes 6/10*5/10 = 3/5*1/2 = 3/10 = 30%
In the 00|0 = 100%
Your tens hitting 50 = 60% as well for the same reason as the 90|0.
However, your ones, unlike the 90|0 set, are also the same probability as your tens.
This is because in this set, the ones are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
As such, 5 is 6/10.
This becomes 6/10*6/10 = 3/5*3/5 = 9/25 = 36%
This is, again, because the total test is two tests whereby the second test is dependent upon the first test (just like two heads in a row on a coin).
This creates a system for either method which looks something like a car's RPMs through gears.
On a note, given the two, I'd rather use 90|0 method as there's a greater chance of 100 than on 00|0 - in 00|0, you have a 1% chance of accomplishing a 100%, whereas on the 90|0 (due to the nature of how it considers the 0 to be a 10 on the ones die) you have a 100% of hitting 100 or lower; a concept not as easily applicable to the 00|0 due to the 00 being re-used as both a 00 for tens and if combined with a 0 from the ones, only then equaling a 100.
Basically, on the 00|0 method, 99 is your 90|0 equivalent "100 or below" in terms of probability.
Plus; you don't have to change the rule of what the 00 refers to dependent on what the ones die is doing.
Either way, however, neither are equal to a percentile die system.
The closest die to a d100 in terms of probability is a d20, which shares the exact same probability curve as a d100, but within a shorter interval (intervals of 5%, rather than intervals of 1%).
Note: graph represents the given value or lower as a two-test system at each value presented, always including the consideration of the ones position.
Due to comments continually being erased, I'll try to help in this answer as best as I can to summarize the issues raised and discussed in the comments.
Some raised concern by pointing out that 1/10*1/10 is exactly equal to 1/100, or 1%.
This is true.
However, with 2d10 or d100, we aren't asking whether or not a single face value will be achieved.
Instead, we are asking if a given value or below will be achieved, which is a very different question.
It is in this form of the question of result that 2d10 radically departs from d100 probabilities.
It was also raised as a point that the above graph doesn't seem too good at representing odds of a given number or below, which is true; it is a general graph which always includes the ones and shows each value's range.
To plot what it looks like for a given value or below, we have to pick a given number and show that value set in both 90|0 format and 00|0 format.
Here is what 50 and 55 as a target number looks like in probabilities on both forms of 2d10 as percentile.
The values on the left are the tens, while the values on the right are the ones.
The ones are only of need for consideration when the tens on the left hits the 10% results respectively.