It's actually not more involved on iOS (unless you're unlucky and have problems with it not working - I've no idea if it's a tiny minority of people who have problems, or if I'm in a tiny minority of people for whom it works, or if it's somewhere in between). But for those of us who it does work for, it's just as simple as on Android.
You can hold down on the media, and after about a second it brings up a menu where one of the options is "Save", you tap this and it will be saved to your camera roll (ie open Photos app to see it). If it's a message with multiple photos/videos, you so the same thing and all of them will be downloaded at the same time.
Or instead of holding down to get the context menu, you can tap once to open full screen view of the media which has the iOS "share" button in the bottom left, which you can use if you either want to just save one thing from a message that had multiple photos/videos, or if instead of saving to the default place - it's the standard iOS share function, so you can choose from "save to photos" (ie a the default like above), or "save to files" (accessible from the iOS file manager or from other apps), or share directly to a different app (like an email client, or an FTP client if you have one installed, or to an app like DropBox, or any other app you have that supports the OS-wide share menu).
TLDR: The help page you linked to, the top part of that iOS section (that makes it seem complicated) is just explaining how to find an overview of all previously shared media for a contact/group, and then download from there, but you don't need to do that to download it if you're already looking at what you want to download in the main chat window. The bottom part on that page is my second option from above, which is basically identical to what you say about how to do it on Android - just 3 touches (press image, press 3 dots on Android or share icon on iOS, then press save image). But it's actually the more complicated way, with a 2 touch option available (hold down on image, then press "save" :)
(p.s. to any Signal devs reading this, if you'd like to offer a free backups subscription in return for me continuing to evangelise, or beta testing on my iPhone... feel free to reach out :P
And, although personally I'm more keen on the future feature or backing up either to iCloud or to my own server, may I make a suggestion that, if paying for you to backup media, I'd prefer to be able to pay for a "family" plan - as I've moved several family members onto Signal and would like to be able to gift them free backups rather than tell them all to start paying. I suspect I'm not the only person who would think an option to share storage with 5 or more family/friends would be worth paying a bit more than your current single-account price.)