calculus - Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic.

But what about line element?

$$ds^2 = dx^2 + dy^2 +dz^2$$

The way I think about this line element is being geometrically constructed from Pythagoras theorem as:

$$\Delta s^2 =\Delta x^2 + \Delta y^2 +\Delta z^2$$
and then we assume that we can 'get' these quantities ($\Delta x$) to be infinitesimally small (as small as we like) and represent as $dx$ instead, right?

Now then lets take line element on a sphere:

$$ds_2 ^2=r^2sin^2(\theta)d\phi^2 + r^2d\theta ^2$$

It is geometrically constructed again using Pythagoras theorem and assuming that sides of a 'triangle' are small:

$$\Delta s_2 ^2 \approx (rsin(\theta)\Delta \phi)^2 + (r\Delta\theta)^2$$

But this approximation never really becomes equality, the smaller the angles the better it works, but still never equality! People just replace $\Delta->d$ and say $ds$ and say it's differential.

I guess my question is this:

when we write something like

$$ds_2 ^2=r^2sin^2(\theta)d\phi^2 + r^2d\theta ^2$$
we actually have in mind that this quantity contains higher order terms, but they will vanish after we parametrise?
I think about parametrisation in a way:
$$\frac{ds_2^2}{dt^2}=r^2sin^2(\theta)\frac{d\phi^2}{dt^2} + r^2\frac{d\theta^2}{dt^2}$$