Yup, context definitely matters. There are some contexts where the fact that any number has n different nth roots is vitally important. There are other contexts where only real numbers are relevant, or where only positive real numbers are relevant, and where therefore any number has only one or two nth roots. And neither of those contexts is wrong. The existence of multiple contexts makes it nigh impossible to make any sweeping, absolute statements.

Another example that you might appreciate: Physicists and other scientists never actually deal with numbers, as mathematicians generally understand them. The quantities that scientists deal with are all actually distributions of numbers. They can often be approximated as numbers with error bars on them, but in some cases even that approximation isn’t any good, and you need a full description of the error distribution. This fact is vitally important to doing good science, and yet, it’s never actually explicitly presented to scientists in our formal education.