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By the work of Gross and Bernard-Matessi, in dimension 3 $T$-duality should be understood as an exchange of positive and negative local model of Lagrangian torus fibrations, at least in its topological sense. Since in dimension 2, all local models of a Lagrangian torus fibration are generic, so we should expect that $X$ and $X^\vee$ should be topologically the same, just like the case of elliptic $K3$ surfaces.

The philosophy of Gross and Bernard-Matessi should be able to extend to certain noncompact Calabi-Yau manifolds. For example $X=K_{\mathbb{P}^2}$, in this case Gross has constructed a special Lagrangian fibration $f$ away from an anticanonical divisor $D$, and the discriminant is a trivalent graph. It should be regarded as a positive Lagrangian fibration in the sense of Bernard-Matessi and the critical locus of the dual Lagrangian fibration $f^\vee$ on the mirror of $K_{\mathbb{P}^2}$ should be topologically the mirror curve in the sense of Hori-Vafa. If we want to include the divisor $D$ in the fibration, then $f$ will no longer be special Lagrangian since the volume form $\Omega$ will be singular along $D$. So if we include $D$ in the fibration, which can also be regarded as $D$ is fibered by Lagrangian tori with lower dimension $f|_D:D\rightarrow\partial B$, then the base will be a manifold with boundary, which differs from the case considered by Gross and Bernard-Matessi a little bit.

Now let's take away the divisor $D$ and consider the simplest case of $\mathbb{C}^2\setminus D$, by the work of Auroux, it's mirror should be $\mathbb{C}^2$ blow up a point, also taking away a anticanonical divisor $D^\vee$. On the mirror there is an explicit special Lagrangian torus fibration constructed by using symplectic reduction, and the affine structure on $B^\vee$ has only one interior singularity. Away from the boundary, it should be the same with the singular affine structure on the base $B$ of the special Lagrangian fibration of $\mathbb{C}^2\setminus D$. But in this case, $X\setminus D$ and $X^\vee\setminus D^\vee$ are not diffeomorphic with each other. Similar phenomenon happens for other ALE spaces given by resolution of $\mathbb{C}^2/\mathbb{Z}_m$, although $X$ and $X^\vee$ are homologically similar in the sense that $H_2(X,\mathbb{Z})\cong H_2(X^\vee,\mathbb{Z})$. Also notice that these spaces are hyperkahler for dimensional reasons, and hyperkahler rotation is expected to produce diffeomorphic mirrors.

Here the divisors $D$ and $D^\vee$ are taken away to avoid the consideration of the superpotentials. If we also include $D$ and $D^\vee$, then $B$ will be diffeomorphic to $\mathbb{R}\times\mathbb{R}_{\geq0}$, and $B^\vee$ is diffeomorphic to $\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0}$, even the affine structures differ from each other.

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I agree with the above answer. In a bit more detail, let E be a smooth conic in $\mathbb{C}^2$, of the form $xy=1$. Then the mirror given by Auroux's construction states that the mirror to $\mathbb{C}^2\setminus E$ is the complement of some divisor $D$ in $\widehat{\mathbb{C}^2}$. Well, if we say that the point is $(0,1)$, the blow up can be written explicitly as the zero locus of $ut_1=(z-1)t_2$ in $\mathbb{C}^2 \times P^1$. The relevant divisor $D$ is then the union of $t_2=0$ and $z=0$. Thus we have something like

$$ uv=z-1 $$

where z is in $\mathbb{C}^*$. This is clearly the same as what we started with. In fact, these two spaces are even algebraically equivalent in this example, which would not be true if you consider the analogous construction for $A_m$ surfaces for $m \geq 1$. You can see this example studied in a lot of detail in the thesis work of Pascaleff.

All examples, following Auroux, Gross-Keel-Hacking, ... suggest that for surfaces, if the surface is "log Calabi-Yau," the slogan that hyperkahler rotation is mirror symmetry does seem to work, at least to some extent (for log CY $A_m$ surfaces as above, the mirror is given by a hyperkahler rotation, in general the mirror will at least be diffeomorphic to the original variety). Otherwise, it needs to be corrected along the lines that you suggest (superpotentials, birational transformations, deformations ...). There seems to be an analogue of this for higher dimensional hyperkahler manifolds, but there aren't as many examples which have been explored in the literature.

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I'm pretty sure they are diffeomorphic. For example in the case of C2 blown up at a point, when you remove the relevant divisor you get C2 minus a conic. This is what appears on the other side too.

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