Is it consistent to define the position of a particle in some frame as a vector or is just an informal representation? Velocity and acceleration can be added up and multiplied by real numbers and still have physical meaning (they live in tangent spaces). But what is the physical meaning of adding a position to another or, in relativity's domain, does adding two events have any sense? If this has no sense, then defining the position of a particle as a "vector living in a vector space" is surely wrong?
It depends on context:
In the context of affine spaces (such as, Newtonian mechanics or SR), positions are strictly speaking not vectors, but position differences/displacements are vectors, i.e. positions measured relative to some chosen origin/fiducial point are vectors.
No, positions are not really vectors in a vector space. They are points in an affine space https://en.m.wikipedia.org/wiki/Affine_space . An affine space comes with a vector space though and you can use the vector space to parametrize the affine space nicely. This, is of course, the classical (and in fact, special relativistic I.e. no general relativity) description.
Historically, vectors were introduced in geometry and physics (typically in mechanics)
Geometry means three dimensional space
A Euclidean vector, is thus an entity endowed with a magnitude (the length of the line segment (A, B)) and a direction (the direction from A to B). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction. For example, velocity, forces and acceleration are represented by vectors.
This includes defining the location of a particle in three dimensional space.
Actually the question should be :
In a general physical sense, can the th position of a particle really a vector?
as assigning numbers to particles is the process of modeling with algebra and mathematics.