Another question on this site asks why we have chosen our number system to be decimal base 10. There are others asking basically the same thing as well.
I'm not really satisfied with any of the answers, because most of the answers given seem to suggest that base 10 was chosen because we have $10$ fingers. However, this would seem to me to imply that we should be using decimal base $11$. Supposing we use the scheme of calling decimal 10 "A", then on our fingers we would could count $1, 2, 3, 4, 5$ on the first hand, and then $6, 7, 8, 9, A,$ on the second. Only then would we be out of fingers and need to roll over to 10 which would be decimal $11$. Likewise, a similar argument could be made for base 6 counting on only one hand, as there are five digits before one runs out and needs to roll over to $10$, in this case for the decimal value 6.
For base 6 the argument could be made that the thumb is not counted, and thus base 5 is more natural, but the fact remains that we don't use base 5 either, we use base 10, and not counting thumbs on either hand would result in us using base 9, not decimal base 10, so I feel like this argument does not hold water either.
An alternate explanation, that base 10 is an abbreviation of base 60 seems slightly more likely, but base 60 seems rather unweildy to being with, which leads me to the question, why don't we simply use base 11, as our 10 fingers seem most suited to it? As far as I am aware no culture has ever widely used it.
The idea of a "base", and even the idea implicit in it of a consistent positional numbering system, is a relatively modern one. For that matter, even "base 10", in the sense of decimal numerals — the Hindu–Arabic numeral system we all use today — is relatively modern; witness the fact that much of Europe didn't begin to use it until well into the second millenium. The idea that we can count in any base is even newer.
Your argument might make sense if humanity started with the idea in mind of using a base-$b$ representation for some $b$, and then looked to their fingers to decide what the base $b$ should be. This of course is not what happened, nor even is it imaginable in any culture. The concrete precedes the abstract.
Instead, what we see historically is counting on one's fingers, and thus counting by fives or by tens, not counting in base 10. If you're counting on two hands, when you reach $10$ you literally run out of fingers to count, and that is where you have to leave a mental (or physical) note to yourself that you're done with one round of counting, and begin counting anew. (Similarly $5$, if using one hand.)
We can see traces of this "counting by $5$" or "counting by $10$" in early systems like Roman numerals: note that $8$ is represented as "VIII", denoting one count of five (done with one hand's worth of counting), and starting again, reaching up to $3$ in the process. Similarly, the representations "XX" and "XXX" show that they were being thought of as "two tens" and "three tens", rather than as "in base $10$, three in the tens place and zero in the units place" — the idea of base $10$ is not actually present. Thus "XXXVIII" literally denotes the process of counting: "three tens (three double-hands), a five (one hand) and three fingers". (There are even traces of counting by $20$s; consider the English "score" and French number names).
It was only after centuries of already counting by $10$s, conducting transactions with $10$-based words for numbers and so on, that the base-$10$ representation arose, as a representation system for the same numbers that everyone was already accustomed to thinking of in tens, and (very slowly) spread across the world.