I think an example is the most clear way I can phrase my question. I'm very iffy on the details, but this is sort of my understanding:
Consider the metric spaces $(\mathbb{R}^2, d_{\mathbb{R}^2})$ and $(\mathbb{C}, d_{\mathbb{C}})$ with the standard metrics. The map $\varphi$ defined by
$$ \begin{align} \varphi\colon (\mathbb{R}^2, d_{\mathbb{R}^2}) &\rightarrow (\mathbb{C}, d_{\mathbb{C}}) \\ (x,y) &\mapsto x + iy \end{align}$$
is a bijective isometry (is "global isometry" the usual term?). Thus, the two spaces are isomorphic as metric spaces, and we may regard them as equivalent as metric spaces. So every result about one which only relies on its metric space structure may immediately be applied to the other.
For instance, a sequence $\{ z_k \}_{k=1}^\infty$, with $z_k = (x_k, y_k)$, in $\mathbb{R}^2$ converges to $z = (x, y)$ if and only if each coordinate sequence $\{ x_k \}_{k=1}^\infty$ and $\{ y_k \}_{k=1}^\infty$ converge to $x$ and $y$ in $\mathbb{R}$ respectively. From the above it follows that the analogous sequence $\{ \tilde{z}_k \}_{k=1}^\infty$, with $\tilde{z}_k = \varphi(z_k) = x_k + iy_k$, in $\mathbb{C}$ converges to $\tilde{z} = \varphi(z) = x + iy$ if and only if the coordinate sequences (which we now interpret as the sequences corresponding to the real and imaginary parts of the complex sequence) converge to the limits mentioned above.
In my analysis course we proved each of the above separately and did not mention that the two spaces were isometric. So I don't actually know if the above is correct. But later it seemed like we were supposed to apply a bunch of result about Euclidean spaces to the complex numbers, though it was never explicitly mentioned. And this answer to another question claims that a finite-dimensional inner product space is always a Hilbert space, since it is isometric to $\mathbb{R}^n$ for some $n \in \mathbb{N}$, and completeness is a property of inner product spaces. So I guess there is something to it. It certainly seems intuitive enough to me.
I was wondering about a few things: How exactly do we know that some property of an object is a metric, topological, vector space, etc. property? Is it simply that we start a proof with "assume that $X$ is a metric space", and then we may assume that whatever result applies to any other metric space isometric to $X$?
Secondly, is there a more formal and perhaps rigorous way to describe this application? Say, the [type of structure] space $X$ has property [property of space], this [property of space] is a [type of structure] property, and $X$ and $Y$ are isomorphic as [type of structure] spaces, so $Y$ also has [property of space]. Is there a way to describe this more formally? (I don't know if it is helpful, but I know a bit of category theory, around the level of Aluffi.)
