This doesn't work if you have to deal with multiplication of numbers that are not integers. You can adjust your idea to work with rational numbers (i.e. ratios of integers) but you will have trouble once you start wanting to multiply irrational numbers like e and pi where you cannot treat multiplication easily as repeated addition.

The actual answer here is that the set of real numbers form a structure called an ordered field and that the nice properties we are familiar with from algebra (for ex that a product of two negatives is positive) can be proved from properties of ordered fields.

Don't confuse the wording "set of real numbers" here, this is just the technical name for the collection of numbers people use from elementary algebra on through to calculus.