I've been reading about base64 conversion, and what I understand is that the encoded version of the original data will be 133% of the original size.
Then, I'm reading about how YouTube is able to have unique identifiers to their videos like FJZQSHn7fc and the reason was: an 11 character base64 string can map to a huge number.
Wait, say a huge number contains 20 characters, then wouldn't a base64 encoded string be 133% of that size, not shorter?
I'm very confused. Are there different types of base64 conversion (string to base64 vs. decimal to base64), once resulting in a bigger, and the other in a smaller resulting string?
You are confusing what things are being compared.
There are 2 statements, both comparing different things:

"base64 encoding is 133% bigger than original size"
"An 11 character base64 string can encode a huge number"

In the case of 1, they are normally referring to a string encoded maybe with ASCII using 8bits a character, and comparing that with the same string encoded in base64. That is 133% bigger, because in base64 you can't use all 255 bit combinations in every byte.
In the case of 2, they are comparing using a numeric identifier, and then either encoding it as base64, or base10. In this case, base64 is a lot shorter than base10.
You can also think of the (1) case as comparing base256 against base64, and the (2) case as comparing base10 against base64.
Base64 conversion decimals