In differential geometry the idea is that you have a manifold and some other quantities or geometrical objects on it that (both the manifold and those quantities) are quite independent of coordinates we use in their description. Therefore by definition, the coordinate transformation do not change the geometry.
But of course, you can define a transformation that changes the manifold (or fields).
Knowledge of both the metric tensor and the tensor of torsion is enough to determine whole geometry of manifold.
The most general transformation will transform coordinates, the metric tensor, the torsion tensor and also all other fields defined on the manifold. In case when metric, torsion and other tensor fields transform accordingly to coordinates the way tensor fields should transform, we have an coordinate transformation and manifold and all ''geometry'' (and physics) is preserved. In case metric and torsion transforms accordingly but other physical quantities does not, you are probably changing physics but preserving manifold. In case torsion or metric transformed differently than they should based on tensor transformation laws, you have defined new manifold.
As far as Poincare transformation goes, to me it looks just like semantics problem. You surely can define something that resembles Poincare transformations on general manifold.
And of course you can define transformations that amounts to boosts and rotations and translations on general manifold, but remember that these kind of ''euclidean'' concepts have some sense, as far as we are concerned in coordinate transformations, only locally.
