There's other goofy stuff people do with df/dx, right? Like in a u-substitution you literally do "algebra" with it.

The solution of differential equations by separation of variables in physics is also notated in an abusive way. You have some differential equation

dy/dx = g(x)h(y)

You separate the variables by some quick manipulations

dy/h(y) = g(x) dx

And then you have a small step in some coordinate on both sides. So by integrating both sides

\int 1/h(y) dy = \int g(x) dx

you find a solution to your differential equation. Obviously there's a real formal procedure underneath it with also some safeguards. For example you're supposed to check that h(y) doesn't equal 0 at any point. But the happy path in physics is often done without worrying about all that.

It's funny that most intro calculus courses will make it a point to remind you that "dy/dx" isn't a fraction, then when they get to integration & diffeqs they want you to forget that and start manipulating them as such. I think most intro courses would be better off skipping everything on convergence tests (which feel really arbitrary anyway until you understand more of complex analysis) and instead use that time better explaining differentials (and maybe a peek into differential forms)

If you thought that was goofy, check out "Umbral calculus" https://en.wikipedia.org/wiki/Umbral_calculus

well.. no, not exactly. If u = u(x) then du = u'(x) dx holds rigorously, and then you can substitute du/u' = dx in an integral.