This is a very hand-wavy way of discerning distros, but they basically differ by 3 things:

1. the set of the defaults they ship with when installed,
2. the system packages that they distribute through their official package manager hosted on their own package repositories, and
3. the package manager, which encompasses the distro’s release strategy.

Major distros generally manage how a package gets built on their distros, in a way that’s compatible with the rest of their package repository, while smaller players may choose to directly use one of the repositories from the major distros, go their own route, or do something in between, i.e. repackage some of the packages from the upstream repositories. Typically, the smaller distros re-use large parts of a larger distro and give a sort of flavour to the larger distro. In the Linux community, these larger distros end up being called “bases”, and many smaller distros are generally “based on” some larger distro.

Manjaro is based on Archlinux, which, incidentally, is also what the newer SteamOS is based on (SteamOS used to be Ubuntu-based). Whether Manjaro actually provides benefits remains to be seen, cause their reputation has been really bad for several years because of how they’ve soured their relationship with a really supportive community earlier on in their life, and badly handled the distribution and communications of several critical packages. I haven’t followed their news in a while, but if they stroke a deal with the company to work together and ship essentially proprietary software or drivers, you can certainly expect some advantage, at least earlier on, but experience tells us that these usually don’t end up well in the long term.

As far as the handheld market goes, you aren’t wrong: every company and their mother that has a potential to get into this market is now ogling at the chance to gain that market share after seeing the success of the Switch and Deck. Many see the Deck as an underpowered machine and believe that they can offer better specs at lower prices (particularly large companies as they typically already have the benefit of economics of scale). AFAIK the Deck has been unbeatable in terms of market share, but that might be outdated info from several months ago.

Manjaro vs. SteamOS, likely no benefit. My speculation is that the company probably wanted to develop their own software but didn't have the technical expertise to develop for Linux. Hence, they partnered with Manjaro. As for why they went with Linux in the first place, it's likely because Windows is a mess and they likely considered that Linux would be a more marketable (and cheaper) operating system.

And yes, these products aren't particularly expanding the market, but it seems that companies would still like to throw their hat into the ring, if only because getting in early means that they'll have a more dedicated fanbase for their later handheld products

Proton is just steam’s wrapper for wine (Apple’s wine wrapper is Apple Game Kit)

You’ll use proton if you’re using Steam

OS differences if any would be better support for non-Steam applications

You should view it as just competing for market share, if both run Linux then performance differences are just hardware