I am wondering why the gcd/lcm of polynomials have to be monic. The idea that constant multiples of it cannot also be gcd/lcms makes sense to me but I am unsure what happens when we evaluate the gcd/lcm of say, $4x^2-1$ and $2x-1$ in the $\mathbb{Z}[x]$ ring.
The intuitive gcd would be $4x^2-1$ and lcm $2x-1$, but they are not monic so they cannot be gcd/lcms by the stricter definition.
The division algorithm doesn't work, in general, when the divisor is not monic in $\mathbb Z[x]$.
In particular, $\mathbb Z[x]$ is not a principal ideal domain, so the "natural" GCD of $2$ and $x$ is the ideal $\langle 2,x\rangle$, which is no a principal ideal, so it is not associated with an element of $\mathbb Z[x]$.
(The term "ideal" was chosen, in fact, because, ideally, there should be a common divisor of two or more element of a number field, and these numbers that "should exist" were considered "ideal divisors" even before they had a formal definition as subsets of the ring.)
You can still define the GCD in any ring with unique factorization, like $\mathbb Z[x]$, but it is a far weaker thing, because the best - or should I say "ideal" - definition of GCD is in terms of the sum of ideals.