Unfortunately it is virtually impossible to identify by year, or even a year range, most civilian-manufactured ammunition.
The "H" headstamp was used on Winchester manufactured rim fire cartridges for well over 100 years, and may still be used.
As someone mentioned, given the rough length of the case (the .44 Short case was actually around .688 inches long), it is possible that the cartridge is actually a .44 Short and not a .44 Henry.
.44 Short was virtually identical in all dimensions to the .44 Henry except case length.
I'm not sure how it would have fed through a Henry rifle's magazine, but it is possible that it would feed without a problem.
The double firing pin strike is a hallmark of the Henry rifle and the later Winchester Model 1866.
"The gift which I am sending you is called a dog, and is in fact the most precious and valuable possession of mankind" -Theodorus Gaza
Baby Jesus cries when the fat redneck doesn't have military-grade firepower.