No, that’s probably not a coincidence. The theoretical model that may explain your discovery that MpRp= 4h/2pic$M_{p} R_{p} = \frac{4 h}{2 \pi c}$, where Mp$M_{p}$ and Rp$R_{p}$ are the proton’s mass and charge radius, is that thethe proton may be composed of four photons each of mass equal to one quarter of the proton’s mass. That’s where the ‘4’$4$ in your equation may come from. The wavelength of a photon of mass equal to one quarter of the proton’s mass can be calculated from L= h/mc$L = \frac{h}{m c}$, and is 5.28563 fm$5.28563 ~ fm$. The circumference of the proton, using its 2018 CODATA radius value, is: 2pi(0.8414 fm) = 5.28667 fm$2 \pi \left( 0.8414 ~ fm \right) = 5.28667 ~ fm$. So, the wavelength of a quarter proton mass photon, and the proton’s circumference calculated from its 2018 CODATA radius value are almost identical. They differ by only 0.02%$0.02 \%$, or 1$1$ part in 5000$5000$. Could the proton be composed of four quarter proton mass photons? To decide, one needs to calculate the volume of a photon of mass equal to a quarter of the proton’s mass and compare that volume to the proton’s volume. Will four such photons fit inside the proton?
Using Maxwell’s model of the shape of a light wave - a sine curve from 0$0$ to 2pi$2 \pi$, with amplitude ‘r’$r$ - the volume of its enclosing cylinder is its wavelength times the area of its base circle of radius ‘r’$r$: V=(2pir)(pir^2) = 2(pi^2)(r^3)$V = \left( 2 \pi r \right) \left( \pi r^{2} \right) = 2 \pi^{2} r^{3}$. That volume is almost 5$5$ times larger than the 3d3D volume of the proton, so the four photons cannot fit in the proton’s 3d3D interior, but it is exactly equal to the surface volume of a 4-sphere of radius equal to the proton’s radius. (The surface volume of a 4-sphere is: S = 2(pi^2)(r^3)$S = 2 \pi^{2} r^{3}$, which is exactly the same formula as the formula for the volume of the photon’s enclosing cylinder.) And, since photons are bosons, all four can occupy the same space at the same time. So, it’s possible the proton is composed of four quarter proton mass photons, all occupying the surface volume of a 4-sphere of radius equal to the proton’s radius, in the 4d4D space surrounding the 3d3D location of the proton. If such is the case, if 100%$100 \%$ of the mass of the proton is found in the 4d4D space surrounding the proton, that means the interior 3d3D volume of the proton is devoid of matter. The proton may be hollow with respect to our 3d3D space, according to this model. It could be that the surface of the proton is not the surface of a 3d3D object, as is currently thought, but the location of the intersection of the proton’s mass (which may be in the surface of a 4-sphere of radius equal to the proton’s radius) with our 3d3D space (aka, the Higgs field). If true, this invalidates the quark theory model of hadron structure, which posits that the 3d3D interior of the proton is where point particle quarks and the strong force are located. So, no. I don’t believe the relationship you found is a coincidence at all. It may be a consequence of the proton having a structure different from what it is currently believed to have.
