The notation “$\mathrm{d}x$” has a number of different meanings in different contexts,
and it can be hard to keep track of them. And in the context of using
a derivative to approximate the value of a function, many presentations
intentionally conflate two different meanings because it makes it
easier to write out the math. I'll try to clear things up by explaining
distinguishing the two usages.
Review of concepts and standard notation:
Suppose we have a differentiable function $f\left(x\right)$ and we
think about the curve/graph given by $y=f\left(x\right)$. Then at
any particular input $a$, we write $f'\left(a\right)$ to mean the
slope of a tangent line to graph of $f$ at $x=a$. Using “$\Delta x$”
as a variable, we have $f'\left(a\right)={\displaystyle \lim_{\Delta x\to0}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x}}$.
It's common to abbreviate something like $f\left(a+\Delta x\right)-f\left(a\right)$
as $\Delta y$ (though we must remember that it depends on both of
the numbers $a$ and $\Delta x$).
With this abbreviation, we have
$f'\left(a\right)={\displaystyle \lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}}$.
Another notation for derivatives can be used to remind us of this:
$f'\left(a\right)=\left.\dfrac{\mathrm{d}y}{\mathrm{d}x}\right|_{x=a}$.
Note that this is just notation: “$\mathrm{d}y$” isn't a number
here or anything.
Linear Approximation:
Anyway, because $f'\left(a\right)={\displaystyle \lim_{\Delta x\to0}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x}}$,
we have a useful approximation:
“$f'\left(a\right)\approx{\displaystyle \frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x}\text{ when }\Delta x}$
is small”. When we know $a,f\left(a\right),f'\left(a\right)$, and
a small number $\Delta x$, we might rewrite this in other forms,
like: “$\Delta y\approx f'\left(a\right)\Delta x$” or “$\dfrac{\Delta y}{\Delta x}\approx\left.\dfrac{\mathrm{d}y}{\mathrm{d}x}\right|_{x=a}$”,
etc.
Discarding $a$:
If we want to imagine all the slopes of tangent lines at once, or
just want to emphasize that $a$ is an $x$-coordinate of interest,
we might sometimes use the variable “$x$” in place of “$a$”
in all of this, with one notational change. So we'd have $f'\left(x\right)={\displaystyle \lim_{\Delta x\to0}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}}$,
$\Delta y=f\left(x+\Delta x\right)-f\left(x\right)$, etc. But instead
of writing something like “$\left.\dfrac{\mathrm{d}y}{\mathrm{d}x}\right|_{x=x}$”,
it's conventional to just write “$\dfrac{\mathrm{d}y}{\mathrm{d}x}$”.
Then we can write nice-looking things like “$\dfrac{\Delta y}{\Delta x}\approx\dfrac{\mathrm{d}y}{\mathrm{d}x}$”
and “$\Delta y\approx\dfrac{\mathrm{d}y}{\mathrm{d}x}\Delta x$”.
A nonstandard notation:
In the above discussion, we had $\Delta y\approx\dfrac{\mathrm{d}y}{\mathrm{d}x}\Delta x$.
Here, $\Delta x$ is a change in inputs, $\Delta y$ is a change in
outputs, and $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ is the limit of a
ratio of changes as the change in inputs gets small. But we don't
have a symbol for the approximate change in outputs: $\dfrac{\mathrm{d}y}{\mathrm{d}x}\Delta x$.
I don't think I've seen the following notation used anywhere, but
by analogy with capital Delta in $\Delta y$, I'm going to define
$\delta y$ to be this quantity $\dfrac{\mathrm{d}y}{\mathrm{d}x}\Delta x$
(which depends on both $x$ and $\Delta x$). Then we have $\Delta y\approx\delta y$,
so that ${\displaystyle \frac{\Delta y}{\Delta x}\approx\frac{\delta y}{\Delta x}=}\dfrac{\mathrm{d}y}{\mathrm{d}x}$.
This still looks a little inelegant, so to make it look nicer, let's
just define $\delta x$ to be $\Delta x$. If you want, you could
think of $\delta y$ and $\delta x$ as approximations of $\Delta y$
and $\Delta x$; it just so happens that $\delta x$ approximates
$\Delta x$ with no error. Then we have the really pretty $\boxed{{\displaystyle \frac{\Delta y}{\Delta x}\approx\frac{\delta y}{\delta x}=\frac{\mathrm{d}y}{\mathrm{d}x}}}$.
The confusing bit:
Now, here's the issue/confusing part. Even though in the discussion
above $\mathrm{d}y$ wasn't a number (just a part of the derivative
notation) and $\delta y$ was a number (the approximate change in
$y$ coming from the derivative), ${\displaystyle \frac{\delta y}{\delta x}=\frac{\mathrm{d}y}{\mathrm{d}x}}$
looks really nice.
It looks so nice that, in this sort of approximation context, it's relatively
common to use “$\mathrm{d}y$”
for the number I have called “$\delta y$”. And then to use “$\mathrm{d}x$”
for the number I called “$\delta x$”, which is just $\Delta x$.
Since the ratio of these numbers equals the derivative, this doesn't
mess up any of the algebra you would like to do with the derivative/these
approximations. This is a little weird because, in other contexts, the notation “$\dfrac{\mathrm{d}y}{\mathrm{d}x}$”
is meant to suggest a limit as $\Delta x$ gets arbitrarily small, not just “small enough so the derivative gives us a good approximation”.
For examples, this $\mathrm{d}y$ as $\delta y$ and $\mathrm{d}x$
as $\delta x$ notation is used in videos like the ones mentioned
in a comment by the OP Yashasv Prajapati:
blackpenredpen's “delta y vs. dy (differential)” and The Math Sorcerer's “How to Compute Delta y and the Differential dy”. For the record, things written like “$\mathrm{d}y$” are called "differentials" whether they're used to represent numbers or other things.
Punchline:
To spell out the source of the confusion in the question:
When blackpenredpen writes “${\displaystyle \lim_{\Delta x\to0}}\dfrac{\Delta y}{\Delta x}=\dfrac{dy}{dx}$”, it's true with the right side as the derivative, but nonsense (or, at best, confusing) with the $dx$ on the right side as representing $\delta x=\Delta x$. But since ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\delta y}{\delta x}}$, it never hurts his calculations to write $\delta x$ as $\mathrm{d}x$, etc.
An aside about other notation:
Not every treatment of these sorts of approximations uses this confusing
convention. Note that the tangent line at $x=a$ is given by the equation
$y=f\left(a\right)+f'\left(a\right)\left(x-a\right)$. This makes
the tangent line the graph of the function $L\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)$.
If we set $\Delta x=x-a$, we have $f\left(x\right)=f\left(a+\Delta x\right)\approx f\left(a\right)+f'\left(a\right)\Delta x=L\left(x\right)$.
For example, the essentially the same material about approximations is written this
way at Paul's Online Notes about linear approximation.
