Given a (smooth) manifold $M$, are there any sufficient, intrinsic properties that would tell you there exists a (smooth) manifold $N$ such that $M$ is diffeomorphic to $TN?$ There are some obvious necessary conditions (like $M$ needs to be even-dimensional), but I'm more interested in whether any sufficient conditions are known.
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.
One necessary set of conditions:
Step 1: you need $M$ to have the homotopy-type of a submanifold $N$ half the dimension of $M$. This forces $M$ to be a vector bundle over $N$ with the fibre the same dimension as the base with some reasonable hypothesis (see texts on the h-cobordism theorem, like Kosinski, where these kinds of theorems are proven).
Step 2: Given a vector bundle, how do you know if it is the tangent bundle of the base space? For this you need an exponential map, or to show the vector bundle is diffeomorphic to a tubular neighbourhood of $\Delta N$ in $N \times N$. Or you could compare the classifying map of $TN$ with that of your vector bundle structure on $M$.