The above approach uses the factor 2.2. Why would your factors be any better?

2.2 is just an arbitrary number.

e is also just an arbitrary number, but it looks like someone calculated it using sophisticated math.

Wouldn't it be 2.4 as you are adding 20% of the already doubled estimate?

Yes.

Nitpick: x * 2 * 1.2 == x * 2.4, not 2.2

Extended nitpick: Technically, they only said "then add 20%". They didn't specify of what that 20% is. It's up to interpretation whether they mean 20% of the original value or the doubled value.

Because e is more scientific.

Semi-seriously, e is a number defined by recursive growth.

So if you estimate to Level 1, and then break down to Level 2 sub-tasks, then Level 3, your estimate will be out by the same amount each time. and your total error will be a multiplicative series.

If you have an infinite number of levels - which is reasonable for most real-world projects, although this may not be obvious in the planning stages - e will get you nicely in the ballpark.