The core of this question probably applies to functions and equations in general but figuring out how this works in polynomials will probably solve the problem as a whole. I have some ideas in my mind as to how polynomials and the functions associated with them relate but I suspect I might be missing something so I will be listing them below to see if anyone will correct something or clarify a point further. Any contribution is appreciated.
(1) When we see an equation of the form: ax + b = cx + d, if we do not want to involve functions or another variable into this at all, we can simply solve this equation as a univariate polynomial equation by arranging the terms as the following: (a - c)x + (b - d) = 0 which will lead us to the solution of this univariate, 1st degree equation which is a value for x.
(2) However, when we are given polynomial expressions inside an equation, we can "choose to view" each of them as seperate polynomial functions of x such that each of them equal y. What I mean by this is:
ax + b = cx + d can be viewed as two polynomial functions, namely P(x) = ax + b and Q(x) = cx + d, each having as a second variable y (here y is the dependent variable, the value of the function).
So instead of looking at this equation as an equation in one variable, we can also think of it as an equation in two variables where the y components cancel anyways since we assume any polynomial to represent a function P(x) = y.
So if we assume that P(x) = y and Q(x) = y, then P(x) - y = 0 and Q(x) - y = 0 and when we equate these two polynomials in two variables this way: P(x) - y = Q (x) - y and arrange the terms as:
P(x) - y - Q(x) + y = 0, the y's cancel and we end up with P(x) - Q(x) = 0 anyways. But this time we have implications for both x and y i.e. any x value we find while solving this equation and y value that corresponds to it determine the intersection point of the graphs of these two seperate polynomial functions on the coordinate plane. Whereas if we simply solved the first equation without considering anything about an implicit y value or functions, we would just end up with a value for x. There is nothing wrong with that mathematically but this other way of looking at the equation gives us another perspective as to how polynomial functions work .