The essential property of anything mathematicians tack the word "number" onto isn't the fact that there's always a "next" number to count with. And it's very us

The essential property of anything mathematicians tack the word "number" onto isn't the fact that there's always a "next" number to count with. And it's very useful to do math with reals, otherwise we wouldn't be able to talk about things like the circumference of a circle, because you need an irrational, transcendental (not the root of a finite rational polynomial) constant. Generally speaking, a good chunk of the number systems mathematicians worked with involve taking some existing number system (tracing back to the natural numbers), finding some operation that takes you outside that system, and then building a new system to cover that hole. A system that has no such holes given a particular operation is said to "close over" that operation. So, integers close over subtraction; rationals close over division, and complex numbers close over exponentiation. Since all of these systems ultimately derived from operations over the natural numbers, it makes sense to also call them numbers, even though they might lack certain symmetries or properties of simpler number systems. Also, the word "measure" is already taken by a different concept.