Yes, https://www.mathsisfun.com/numbers/bases.html has a bunch more examples to show why the base of the number system is always represented by 10, because 10 is a short hand we use for d1*b^1 + d2*b^0 where d is a digit between 0 and base-1, and b is the base. b^0 is always one and represents the first digit at the first position. b^1 is the base, so 1*b^1 = the base. And since 10 is 1*b^1 + 0*b^0 it represents the base in any number base system.

Another way to show the same thing with counting:

Base 10: 0, 1, 2, ..., 8, 9, 10, 11, 12...

Base 4: 0, 1, 2, 3, 10, 11, 12, 13, 20, ...

Any base: d1*b^0 , d2*b^0 , ..., d(b-1)*b^0 , d1*b^1 + d1*b^0 , d1*b^1 + d2*b^0 , ...

We assume a 1 in the 10's place and a 0 in the 1's represents 1,2,3,...,10 of something instead of 0,1,2,3,10 of something because from our perspective we learned numbers in base 10 with 9 digits, but the alien learned 10 means 4 of something in base 4.

Assuming they use the same numbers then yes, 10 is 10 to us but it represents the separator of orders of magnitude. 10 in base 4 would from our perspective be 4, but ignoring the specific numbers and adapting them for the sake of the joke, the alien would write it as 10 and "base 10" would just be base 4. It's like how hexadecimal uses 0-9 and a-f to represent 16 numbers even though we don't have that many in decimal systems. The joke is a translation issue. I feel like I'm over explaining now though...

Yes