In, say, a linear regression problem, why does the dot product of the weight vector (w) and feature vector (X) result in the value of the dependent variable (y) i.e., why does addition of products of weights and features result in y? Could we have chosen an operator other than addition, to act upon the product of the weights? I understand the intuition behind y=mx for a single variable, but why do we choose to addition as the operator in case of multiple features?
There's no need to use summation, but we choose it because it has several nice properties. With summation, we are dealing with linear functions (hence the name), and we have many nice linear algebra tricks for working with such functions, including closed-form solution using ordinary least squares. It leads to additive effect of the predictors on the target variable, that is easily interpretable and intuitive.
Moreover, it can easily become equivalent to other algebraic operations: if weights have negative signs, it is the same as we subtracted parameters multiplied by positive weights $a + (-w)b = a -wb$, and if we log-transform the variables, then summation becomes multiplication $\log(ab)=\log(a) + \log(b)$.
If you need a functional form of model that does not fit the linear regression framework, you can use nonlinear regression, but it is significantly more complicated to use.