What exactly is a 'dummy variable'?

I was watching an online course (The Calculus You Need - MIT OpenCourseWare), when (around 03:08), the lecturer (Gilbert Strang) says that he doesn't "care what that dummy variable is" (the variable x associated with the function y). He made the following change in the video:

$$\frac{d}{dx}\int_{0}^{x}y(x) = y(x)\Rightarrow \text{ change } \Rightarrow\frac{d}{dx}\int_{0}^{x}y(t)dt = y(x)$$

I don't know why the notation of that variable doesn't cause any ambiguity here (I'm assuming Strang wanted to mean that), once x is clearly creating a dependency relation with the bound and the variable associated with the function y.

Ultimately... why it doesn't make a difference to call the variable associated with y by x (in this case)? What exactly is a 'dummy variable'?

You have asked two questions, one about whether the name of a variable matters, the other about dummy variables. They're different but related.

Strictly speaking, it does not matter what you name your variables. If you have a function $f$ from, say, $\mathbb{R}$ to itself then the elements in both the domain and the codomain are real numbers. You may often see this described as "$y = f(x)$". That usually does no damage, and sometimes helps keep the meaning of numbers straight - those called "$x$" are in the domain while those called "$y$" are in the codomain. But that's not at all required and is sometimes confusing. You should talk about the function $\sin$, not the function $\sin(x)$.

Now for dummy variables. The expression $$ \sum_{n=1}^3 n^2 $$ means just $$ 1^2 + 2^2 +3^2 . $$ There is no "$n$" in it. You could write the same thing as $$ \sum_{@=1}^3 @^2 . $$ The $n$ and the $@$ are dummy variables - they are not there. The same is true for the "variable of integration": $$ \int_1^2 \sin(x) dx = \int_1^2 \sin(@) d@ = \int_1^2 \sin . $$ The last of these is unambiguous when you work with $\sin$ as a real valued function of a real variable.

That said, there are times when a (dummy) variable of integration is helpful. With it you can distinguish between $$ \int_1^2 \sin(xt^2) dx \text{ and } \int_1^2 \sin(xt^2) dt . $$ The first is a number that depends on the value of $t$, the second a number that depends on the value of $x$.

The dummy variable $x$ and the $dx$ are quite important when you think about applications of integrals in geometry and physics. If you imagine the integral of $\sin$ as calculating the area under the sine curve then the expression $$ \sin(x) dx $$ is the (infinitesimal) area of a rectangle with height $\sin(x)$ and (infinitesimal) base $dx$. Each of those quantities has units length and their product has units area. The integral sign is Leibniz's elongated "S", for "sum".

To add to what was already said, I find it helpful to think in programming terms. What is a dummy variable? It's a local variable with very limited scope - a sum, integral, or other similar construct - and it had better not appear anywhere outside that. If it does, as in that initial expression $\frac{d}{dx}\int_0^x y(x)$, we'll get errors and unpredictable results. For example, here? In order to get something that parses, we have to take that integral with respect to some other variable - which means that inside the integral, $x$ and $y(x)$ are constants. We then get $\int_0^x y(x)\,dt=xy(x)$, so its derivative with respect to $x$ is $y(x)+xy'(x)$, and the solution of the equation is for $y(x)$ to be a constant function.
The equation after the change is clearly what was intended all along, and it doesn't mean the same thing as the equation before the change. It doesn't matter what we call the dummy variable - as long as we're careful not to conflict it with a variable name on the outside and cause out-of-scope errors.

As for leaving the $d?$ off the integral? I'm not a fan of that; it's a delimiter, which clearly marks what we're integrating, and what variable we're integrating with respect to. As far as I'm concerned, it's part of the integral sign. The only way I'd ever leave it off is if the dummy variable is left out entirely. $\int_a^b f(x)\,dx$ or $\int_a^b f$ are OK and unambiguous; $\int_a^b f(x)$ is malformed.