Can we speak of an absolute voltage without a convention?

Yes: if one is attempting to do electromagnetism in 1D then the force becomes constant with distance and it makes sense that $U = k~x$ for some $k.$ This may seem artificial because we are located in 3D space, but it corresponds to an infinite wall of uniform charge, which can be used to describe what things would look like if one were shrunk down and found one's self inside a parallel-plate capacitor. The natural zero would be at the plane of the uniform charge.

Similarly if one is thinking about a point charge next to an infinite conductor, the fact that an infinite conductor must be at constant voltage leads to a natural desire to set this voltage equal to zero. One can then fully enforce this using the method of image charges.

In 2D rather than 3D, one finds a $1/r$ force law which must admit a potential $U_0~ln(r/r_0)$ and this cannot be normalized to be zero at infinity or at the place where the charge is located. This corresponds to a geometry of a charged infinite wire, and one just has to arbitrarily choose a distance $r_0$ away from the infinite wire where the charge will be zero.