Biggest reason is that mixing army and navy is like mixing matter and anti-matter. Don't want to be around when that happens.
In all seriousness, think about what you are asking. Even if it were permitted, typically the training periods are the same, so you could not be in two places at the same time. Then there is the conflict between chains of command, what happens when you receive a MCpl appointment from your Army CoC, but are still only an LS in the Navy, and are not qualified for MS because you were sent your Army leadership course before your Navy trade course had openings?
It is not workable, and only results in chaos. Hence the Matter / Anti-matter analogy.
Edited to correct a naval rank oopsie. :facepalm: