What allows us divide/multiply dx in calculus? I've read nearly all of the threads on this topic but none seem to answer my question or lead me in the best direction.
When performing U-substituion or even in it's most basic form:
$y = 2x$, $dy=2\,dx$.
What allows us to do this and what/where does in come in a calculus textbook? None of my teachers have been able to explain it to me. It makes it difficult for me to proceed when even the basics don't seem intuitive.
I've read that the idea of differentials varies but is there a single understanding regarding them to which I can pertain? I feel I'm missing something by just doing it in my calculations and not understanding why. I've genuinely read all posts regarding this, it's been something on my mind for a long time so please don't close this.
Cheers
The chain rule says $\dfrac d {dx} f(g(x)) = f'(g(x)) g'(x)$.  Therefore

$$
\underbrace{\int f'(g(x)) g'(x)\,dx = \int \left( \frac d {dx} f(g(x)) \right)\,dx}_\text{chain rule} = f(g(x)) + C.
$$

If one abbreviates $g(x)$ as $u$ and $g'(x)\,dx$ as $du$, one writes this as
$$
\int f'(u)\,du = f(u) + C.
$$

That usage is consistent with writing $\dfrac {du} {dx} = g'(x)$, as you learned to do earlier.

If your instructor couldn't tell you that this is about the chain rule (just as integration by parts is about the product rule) then either your instructor either doesn't know the material well or didn't understand what you were asking.
You can understand calculus very well without ever manipulating differentials at all.  In fact, in a rigorous treatment of calculus (such as in Spivak's book Calculus), differentials like $dx$ have been banished (except for lingering in the notation).  To understand $u$-substitution, just use the chain rule in reverse.  $\int f(g(x)) g'(x) \, dx = F(g(x)) + C$, where $F$ is an antiderivative of $f$.  You can check this by taking the derivative of the right hand side, using the chain rule.