Here's a (perhaps messy) formalisation of my problem. I'm not sure if it works, but I'd be glad to explain it further.
Take four integers $a,b,c,d$.
Suppose $\gcd(a,b)=1$ and $\gcd(c,d)>1$.
For some limit $\ell$, let $M_{\ell,a,b}$ be the set of common multiples of $a$ and $b$ smaller than $\ell$, i.e. $M_{\ell,a,b} = \{m \in \mathbb{N} \mid m < \ell, a|m, b|m \}$.
My hypothesis is: if $\ell$ is large enough, $a$ and $b$ will have more common multiples if they have a common factor over 1, than if they don't have one. In other words, for some $\ell \in \mathbb{N}$, if $\gcd(a,b)=1$ and $\gcd(c,d)>1$, then $|M_{\ell,a,b}| > |M_{\ell,c,d}|$ is always true.
Is my hypothesis correct? How do I prove it? Can it be generalised to larger sets of numbers (e.g. what if we compare sets of three numbers)?