Suppose $M$ is an open $2n$-manifold that is homotopy equivalent to a closed smooth $n$-manifold $N$, and suppose $n>2$. Then Haefliger's embedding theorem ensures that the homotopy equivalence $N\to M$ is homotopic to a smooth embedding. Moreover, by Siebenmann's open collar recognition theorem $M$ is diffeomorphic to the normal bundle to this embedding if and only if $M$ is the interior of a compact manifold with boundary such that the inclusion of the boundary induces an isomorphism on the fundamental group. Now it remains to check whether the normal bundle and tangent bundle to the embedding are isomorphic, which of course rarely happens.

A good example is when $N$ is a 3-dimensional lens space and $M=N\times \mathbb R^3$, which is precisely $TN$. By above arguments, any two homotopy equivalent lens spaces have diffeomorphic tangent bundles.

The case $n=2$ seems more delicate.

Suppose $M$ is an open $2n$-manifold that is homotopy equivalent to a closed smooth $n$-manifold $N$, and suppose $n>2$. Then Haefliger's embedding theorem ensures that the homotopy equivalence $N\to M$ is homotopic to a smooth embedding. Moreover, by Siebenmann's open collar recognition theorem $M$ is diffeomorphic to the normal bundle to this embedding if and only if $M$ is the interior of a compact manifold with boundary such that the inclusion of the boundary induces an isomorphism on the fundamental group. Now it remains to check whether the normal bundle and tangent bundle to the embedding are isomorphic, which of course rarely happens.

A good example is when $N$ is an orientable $3$-manifold and $M=N\times \mathbb R^3$, which is precisely $TN$ because orientable $3$-manifolds are parallelizable. By above arguments, any two homotopy equivalent orientable $3$-manifolds have diffeomorphic tangent bundles. Specific examples can be found among lens spaces, such as $L(7,1)$ and $L(7,2)$.

The case $n=2$ seems more delicate.