Forcing homeomorphism
FURTHER EDIT: 35093731895230467514051 raises a valuable point in the comments - do we want our space to "gain points" when we move to a generic extension?
The ur-example of this is the usual space of real numbers. If $V\subseteq W$ are models of set theory with the same $\omega$, then there are two ways to think of "$W$'s version of $\mathbb{R}^V$" - namely, as $\mathbb{R}^V$ with the topology induced by the set $\{$open sets of reals$\}^V$, or as $\mathbb{R}^W$ with the usual topology.
My answer below focuses on the former. There are certainly interesting results to be gotten in this direction; see e.g. this paper by Kunen. However, one might also be interested in the latter (and indeed the latter approach seems probably more interesting in general). Some interesting difficulties can crop here. If I have a definition of a space, $\varphi$, then I can talk about $\varphi^W$ without a problem. However, two different definitions of the same space in $V$ might not define the same space in $W$, so this ultimately is really about the definitions rather than the specific spaces themselves. Somehow, we want a way to take a (reasonable) space and get a "canonical definition" (or something like that) for it, which we can then use to get a notion of what that space should be in a different model.
For this direction, I suggest this paper of Zapletal. (That paper isn't really about generating canonical definitions in any sense, so you might view my focus on definitions above as a bit of a red herring; still, I think that at some level there is a notion of "canonical definition" kicking around, it's just that that notion is much more structural - how the space in question fits into an appropriate category of spaces.)
EDIT: an important distinction between the spaces and structures contexts is: while a structure remains a structure after forcing, a space need not remain a space. This is because forcing may introduce new families of open sets, whose union was not accounted for in the ground model.
Of course, the old topology will form a subbase for a new topology. For that reason it's probably better to talk about spaces in terms of their subbases; I'll just ignore this point below, since it won't affect anything.
An even simpler example of potential but not actual homeomorphism: if $X, Y$ are infinite sets with different cardinalities, then their discrete (or indiscrete) topologies are potentially homeomorphic but not actually homeomorphic. This exactly parallels the non-topological situation.
The interesting part, of course, is characterizing potential homeomorphism. You mention $\mathcal{L}_{\infty\omega}$-equivalence of structures as sufficient for isomorphism; it is in fact necessary as well, so we have a good characterization of potential isomorphism. Another characterization is given by the infinite-length back-and-forth game. So it's reasonable to hope for a good characterization(s) of potential homeomorphism.
It turns out that we can find such characterizations, which parallel the structures case:
The first is a game characterization. It's ultimately subsumed by the next section, but I think it's interesting to think about on its own. Given spaces $\mathcal{X}=(X, \tau)$ and $\mathcal{Y}=(Y, \sigma)$ we let $G_{ph}(\mathcal{X},\mathcal{Y})$ be the following game:
On their move, a given player will play either an element of $X$, an element of $\tau$, an element of $Y$, or an element of $\sigma$.
What sort of thing player $2$ plays is determined by what sort of thing player $1$ plays:
If player $1$ plays an element of $X$ (resp. $Y$) then player $2$'s next move will be an element of $Y$ (resp. $X$).
If player $1$ plays an element of $\tau$ (resp. $\sigma$) then player $2$'s next move will be an element of $\sigma$ (resp. $\tau$).
Player $2$ must play "bijectively" on points. E.g. suppose Player $2$ plays $y\in Y$ in response to Player $1$'s playing $x\in X$; then if Player $1$ later plays $x$ again Player $2$ has to respond with $y$, and if Player $1$ later plays $y$ then Player $2$ has to respond with $x$.
Player $2$'s "set moves" restrict their "point moves": e.g. if Player $2$ plays $V\in\sigma$ in response to player $1$'s play of $U\in\tau$, then they've committed to responding with a point in $V$ whenever Player $1$plays a point in $U$.
(I'm leaving it as an exercise to fully write out the details of this game, but the idea should be clear.)
Say that Player $1$ wins a play of the game if Player $2$ eventually makes an illegal move, and Player $2$ wins otherwise. It's now easy to show that for countable spaces (EDIT: that is, countable set of points and countable set of opens), Player $2$ winning is equivalent to homeomorphism, and this same line of thought translates to:
$\mathcal{X}$ and $\mathcal{Y}$ are potentially homeomorphic iff Player $2$ has a winning strategy in $G_{ph}(\mathcal{X},\mathcal{Y})$.
(Note that $G_{ph}$-games are open games, hence always determined in ZFC, so there's no determinacy issue here. We are using choice, though; if we want to work in ZF alone we need to talk about quasistrategies.)
The second characterization of potential homeomorphism - or rather, class of characterizations - can be given by translating directly into the language of structures. Any topological space $\mathcal{X}=(X,\tau)$ has an associated first-order structure $S_\mathcal{X}$:
$S_\mathcal{X}$ has two sorts, a "points" sort and an "opens" sort; the elements of the points sort are exactly the elements of $X$, and the elements of the opens sort are exactly the elements of $\tau$.
The language of $S_\mathcal{X}$, besides a binary relation, include the binary relation symbol "$\in$" which is interpreted in the obvious way.
It's easy to check that $\mathcal{X}$ and $\mathcal{Y}$ are homeomorphic iff $S_\mathcal{X}$ and $S_\mathcal{Y}$ are isomorphic; so we can now "port over" the characterizations of potential isomorphism of structures. In particular, note that the game characterization above is the "pullback" of the usual game characterization of potential isomorphism along the construction $\mathcal{X}\mapsto S_\mathcal{X}$.
This sort of translation is an instance of a useful general phenomenon: while first-order theories are quite weak, first-order structures are quite expressive. Note that of course the class of structures of the form $S_\mathcal{X}$ for some topological space $\mathcal{X}$ is definitely not first-order definable! This old question of mine contains a similar use of this idea.
EDIT: It's worth mentioning that in the structures case, a lot of interesting stuff happens when you restrict the forcing notions you're using. The Levy collapse $Col(\omega, A\cup B)$ is universal for determining whether $A$ and $B$ are potentially isomorphic: making everything relevant countable solves all possible problems (I have that slogan on a t-shirt). However, we can have stronger notions of potential isomorphism like "become isomorphic in a c.c.c. forcing extension," and here the situation is more complex.
This was studied in particular from a model-theoretic perspective by Baldwin, Laskowski, and Shelah in their papers "Forcing isomorphism" and (sans Baldwin) "Forcing isomorphism II".
I have no idea how similar variations work in the context of topological spaces, but it seems potentially quite interesting. The $\mathcal{X}\mapsto S_\mathcal{X}$ construction suggests that it won't yield a picture which is too new from that of the structures context, but there might still be cool stuff.