Yes, mastering Savage XR will give you quite an advantage in other action/HnS and FPS games.
The key difference between modern FPS (war simulations) and Savage XR is that XR isn't reduced to headshots with insta-projectiles for all weapons, i.e. games like Savage require something that's worth being called a "combat system".
In FPS mode you have weapons without aiming restrictions - that support skill in any situation and don't require camping for maximal precision, just for the sake of realism (i.e. you precicley aim at a nomad being 300 m away while flying in the air with 300 km/h). Compared to modern FPS where weapons just differ in damage, accuracy, fire speed and aiming restriction, Savage XR has additionally different projectile ballistics, projectile speed, splash damage, projectile life time. For surge/flux/blaze/pulse you need tracking skills (since it doesn't kill instantly), for discharger, crossbow, frostbolts, scattergun, coil, tempest, bow etc you need good timing/prediction and aim since the fire rate is so low.
Tracking skills are hard to improve and not really needed anymore in contemporary shooters. Weapon timing/prediction isn't intensively needed either since most weapons in modern shooters have a high fire-rate and insta-projectiles. If you play TF2, UT, Quake, the Savage XR FPS mode will certainly help you to improve quite a lot. For the other FPS games you'll improve just the aim, but you can do that in any shooter.
Secondly Savage XR has a combat system that is all about timing, positioning, reaction, prediction, precision/speed aim, health/stamina management weapon/item switch and pattern analysis. Which is hard to master and mostly specific. You will be able to use the experience in timing, reaction and mouse work in other Hack and Slash games too - unfortunately there are not many decent alternatives though. Timing, reaction, precisio,n weapon/item switch, health management and speed aim will be useful for shooters too.
Thirdly there is the commanding and teamwork part which requires a lot of experience and intuition. The mayority of that experience can't be used for other games.
In a nutshell: Savage XR is a hard and unforgiving game. While being experienced at other games may help you to improve faster at Savage XR, you'll still have a hard time mastering the game, since the game requires a diversity of skills - which are usually quite unique. Being good at Savage XR on the other hand will help you to improve a lot on pretty much any action/FPS game due to the diversitiy of skill requirements. Savage XR has everything.