Does photon have size measurement because of its particle nature like electron's 3.86*10^-13m etc..
When one reaches energies of GeV and experimentally measures interactions that happen below 1 fermi, the nuclear dimensions, the question of size of an elementary particle becomes separated from the concept of its dimensions.
The elementary particles of the standard model have dimensions 0. This certainty comes because the theoretical standard model fits very well practically all available particle data, and the zero dimension of its constituent particles is one of the basic blocs in the computations.
But, at the level below Fermi, elementary zero dimensional particles have complicated interactions, described by form factors, which create a size for them. In particular for the photon the size is seen as a photon structure function, which changes with the interaction exchange energies. It gives an effective size to the photons, not a fixed one, but one depending on the energy of the probing particle.
Form factors are an experimental measurement of these complicated virtual diagrams that exist whenever one elementary particle interacts with another.
So an elementary particle, including the photon, has a size measurement due to the particular interaction possibilities when scattering or being scattered by other elementary particles. A variable size measurement which depends on the energy transfers of the collisions. It is still of dimension zero.
If you have difficulty visualizing this, think of a spark gap, where an electric field is applied. The higher the field the larger the spark seen, the larger the spatial dimensions it occupies, nevertheless the geometry of the gap is fixed. In the same way the geometrical dimensions of the particle are zero, but depending on the energy of interaction ,it has a size.