This is fairly universal, not just the X-H1. All cameras that I know of have either sRGB or even sub-sRGB screens (both viewfinder and rear screen). When you're looking at the feed composing the image, it's fine, since the viewfinder feed is sent in the colorspace the screens can handle (I'm not sure if it's sRGB or a camera colorspace, but, whatever it is, it's small). When you're reviewing an Adobe RGB image on the rear screen, it's sent in Adobe RGB, which the screen can't handle. It's exactly the same as displaying an Adobe RGB image on a normal (non-Adobe RGB) monitor with neither the operating system nor the application being color management aware.
You can't reproduce the effect on a Mac easily, because MacOS is internally color-managed and has been for a long time. It displays all images using the monitor profile. If your monitor is from Apple, even if you've never color-managed it, it has a manufacturer default profile (and most Macs have built-in screens). If your monitor is non-Apple, there are some other default profiles (not sure if they're built in to the OS, or if it downloads them) - and I assume it picks sRGB if nothing else applies. If you've calibrated your monitor, it (of course) uses the calibrated profile.
Newer Windows more or less acts like a Mac - every output device, including monitors, has a profile, and the OS converts images so they look right on the monitor. Old Windows acted like your camera - if you displayed an Adobe RGB image on a non-Adobe RGB monitor, it didn't look right.
The solution to this, and it isn't perfect, is to always set camera colorspaces to sRGB and shoot raw + jpeg. Raw images don't have a profile, so you'll still have the full raw data (raw converters work in very large color spaces, then you pick the export space, which can be sRGB, Adobe and sometimes others). Since the jpeg is sRGB, it looks righ on the rear screen (and on phones, if you use the iOS or Android app). The reason this may not be ideal is if you're trying to heavily edit the jpeg or print the jpeg on a wide-gamut printer. Most of the time, you want the jpeg to be in sRGB, anyway - it's set up for immediate sharing. If you want to do something more than that with the image, at least I am always working with the raw file...