I don't think this is military at all. I suspect it is simply a toolbox made by a tradesman, possibly used by a carpenter as it could hold a decent length saw.
If for a moment you take away the door handle, hasp and staple, and catches, what evidence is there that the box ever had any other means of being secured or carried? I can't see any holes or indications that anything has been removed, so I think it was made originally with all the fittings it now has. The timber is probably floor board.
The two runners on the lid I guess are strengthening battons to stop the lid splitting. If military I would have though there would have been two similar battons on the base at different spacing to allow boxes to interlock and be more stable when stacked.
A military box might have had some metal banding.
Final observation, the sides are nailed onto the ends with floor board nails, as evidenced by the rectangular holes made by their heads. I don't recall seeing these used on military boxes, only screws or round head wire nails.