An event is a 4D point in spacetime. At every point in spacetime there is a tangent space. 4-vectors live in the tangent space. One can contract two events using a metric tensor. Is there a process that moves the event as a point in the manifold to the tangent space?
On a manifold, there is a separate tangent space around every point. The vectors in the tangent space consist of a direction and a magnitude, but they are keyed to the point in question. You can't, in general, move them to other points without assuming special properties of the manifold.
The tangent bundle of a manifold consists of all pairs with a point and a tangent vector in the tangent space of that point. (The tangent bundle is also a manifold, but larger than either the original manifold or the tangent spaces.) See for example Wikipedia: https://en.wikipedia.org/wiki/Tangent_bundle
Consider a point particle with speed. The particle resides on the manifold, its speed on the tangent space at the point, and the pair (point, speed) on the tangent bundle.
The speed and the location of the particle are separate concepts, even though the speed only makes sense at a given point. The tangent bundle combines the information.