(I'd like to premise that I'm not an expert about these topics (just a student), so many of my doubts and perplexities are probably symptoms of my mathematical immaturity)

I'm currently studying differentiable stacks and I'm a little confused by the following statement:

Many classes of interesting geometric objects (orbit spaces, orbifolds, differentiable spaces,...) can be seen as differentiable stacks

This statement fascinated me, because I thought that its meaning was the following

Let $\mathscr{C}$ be a class of geometric objects (orbit spaces, orbifolds, differentiable spaces,...) and let's suppose that we have a natural notion of isomorphism of objects of $\mathscr{C}$. We associate to each object $C\in \mathscr{C}$ a Lie groupoid $\mathcal{G}(C)$ so that: $$C_1\cong C_2 \iff \mathcal{G}(C_1),\mathcal{G}(C_2) \text{ are Morita equivalent }$$ for any $C_1,C_2\in \mathscr{C}$.

This felt natural to me and (in good faith) I took for granted it was like this. But as I went on studying, I realized that probably I am wrong. Let's take the example of orbit spaces.

Let $M$ be a smooth manifold equipped with a left action of a Lie group $G$. Clearly the information of the action is completely encoded in the action groupoid $G\ltimes M$.

But why Morita equivalence should encode a natural notion of isomorphism?

I'm aware that the Morita equivalence of the action groupoids is equivalent to the homeomorphism of the orbit spaces and the isomorphism of the normal representations of the action groupoids.

But in a way this just moves the problem to why "homeomorphism of the orbit spaces and isomorphism of the normal representations of the action groupoids" is the correct notion of isomorphism between spaces of the type $M/G, N/H$.

It's like we are proceding backwards: first we presuppose that Morita invariance is the correct notion of equivalence and then whatever the Morita equivalence implies it's the correct notion of isomorphism between objects of $\mathscr{C}$.

This feels a bit unnatural to me. If I were forced to invent by myself a notion of isomorphism between $M/G,N/H$, I'd probably say that it's a homeomorphism $F:M/G\to N/H$ such that $F, F^{-1}$ both lift to smooth maps $M\to N,N\to M$. I'm NOT saying that this is the smartest approach: I'm just saying that this is what feels more natural to my (mathematically immature) mind.

Thank you in advance for your help!