In one sense, this should not be a coincidence, because the mass and the charge radius are actually determined by more fundamental quantities (quark masses, strong force coupling constant). So given that those quantities have their particular values, in theory it inexorably determines what "mass times charge radius" is going to be.
What should be a coincidence is the appearance of the number 4, when "mass times charge radius" is expressed in natural units. However, since charge radius depends on the distribution of charge inside the proton, it's not inconceivable that, e.g., the distribution of "partons" (quarks and gluons), the distribution of momentum among them, etc., follows some principle which really does imply that the answer is "4 + small corrections". At least, I can't think of a simple reason this should not be so.
For me the deepest fact is that fundamental theory should predict something for this quantity. That prediction might be 4, or it might be e^1.37, but it must be something close to 4.
Update: A comment in chat by @Rococo allows @dandb's observation to be expressed in a very crisp way:
The charge radius of the proton (in muonic hydrogen) is almost exactly four times the reduced Compton wavelength of the proton.
Update 2017: Via P.R. Silva (eqn 6), I have run across a heuristic model of the nucleon in which M = 4/R (in natural units). Here R is the radius of the bag in the "bag model". See Xiangdong Ji, "Mass of the hadron", slide 20. I have not found where this argument originates, but a remark in a 1994 paper by Ji (see paragraph beginning "In the chiral limit...", on the final page) hints at it.