Unfortunately, I havent found a direct answer to your question anywhere so I decided to start trying to give it a try myself. It looks like Opera and Chrome do not accept the Peername extension so the only mobile app that lets you install it is Firefox (haven't tried Focus iet but I guess it doesn't work over there). However, I can't find anywhere a ".eth" website to test it out?
I currently have the Firefox Peercoin Add-On installed on my phone but no domains for testing purposes. I've seen that Vitalik has changed his Twitter name to "vitalik.eth" but apparently it does not establish connection to any website if I try it.
I've found 1 site ending on ".eth" a long time ago. But I can't recall the domain name to access it again. What I do know is that it was possible to browse the site using the "Peername" extension on Chrome easily. I believe that the ".eth" domain name was linked to a URL like it's the case with traditional "TLDs" like ".com", ".net", etc. As of now, people are using Ethereum's ENS to link an ".eth" domain name with an ETH public address. This allows anyone to easily pay you in ETH by typing in your purchased domain name. But I believe its use cases can be expanded far beyond that of finance.
I wonder if we ever get to see webmasters interested in using ".eth" domains sometime in the future? After all, Ethereum is still facing scalability issues (which makes it slow and costly for mainstream use). For testing/experimentation, I'd like to know how to browse ".eth" sites on a mobile device. So far, the only solution seems to be installing the Peername extension on Firefox. I haven't tried this before, but if it works for other Blockchain-based domain names (like ".bit", and ".nxt") it'll be a blast to use. In this regard, the mobile version of Firefox is way ahead of its time since it's not possible to install extensions on the mobile version of Chrome yet.
Most recently, I've found an Ethereum wallet and dApp browser for mobile devices called "Cipher". If it manages to resolve ".eth" addresses in the future, it could become the gateway to the decentralized web on the go. Until the time comes for websites to use domain names ending in ".eth", it's only use case will be for sending payments on the ETH blockchain. I believe that the "vitalik.eth" domain name points to an specific ETH public address on the Blockchain. With ENS improving over time, I hope to see a vast number of sites using decentralized domains for their own benefit.