There is a tangent space for each point of the manifold and thus a nearby manifold point has its own tangent space. If the points are very near we are tempted to identify vectors in the two tangent spaces just as we are tempted to compare wind velocity in two nearby cities, but when the cities are far apart it is not clear what it means to say the wind velocity is the same. Saying it is to the North at 15 km/hr in both cities may be meaningful, but only relative to the Earth’s conventional coordinate system. Manifolds generally lack such special coordinate systems. The mathematical concept of tangent bundles makes one topology of all vectors in all of these tangent spaces.
A manifold may carry an affine connection by which vectors may be carried unaltered along some path thru the manifold. With an affine connection, a path between two points of the manifold establishes a linear transformation between the tangent spaces at those points.
A metric determines a symmetric affine connection, and a symmetric affine connection together with a sphere at some point of the manifold, determines a metric.