There’s an inherent relationship between e and compound interest other have explained.

There’s also a consistent ease of use using exponential/log rules. (1.01)^100 = e^ln(1.01)100 which is in line with the standard formula e^kt for compound interest even though you’re not compounding continuously.

Finally, for frequent long term compounding, like your example, the answers are approximately the same. Taking your interest rate as .01, you answer 1.01^100 = 2.7048 ~ e^(.01x100)= e

I’m sure there are better reasons from specialists , but these are just some observations from teaching the intro to this topic a few times.