I know you can already do this for the Kindle and between Android devices with say Moon+ reader installed. What I meant is for EPUBs and between platforms. I would like to read on my Android phone for example and then pick up where I left off on my desktop. Is this possible?
I'm looking for a solution that does not involve using Kobo or Amazon's apps.
As a general principle, no. But most software vendors provide compatible apps for PC/Mac/Android. So if you will be happy using their reader app, you just install their apps and get synchronization across all platforms.
I had to switch to Bluefire from Kindle after getting a windows 8 tablet since the win8 kindle app doesn't allow sideloaded books. Bluefire on the other hand has apps for ALL the major ecosystems (ios, android, windows). In order to sync between devices you will need to set up a free adobe ID which you will use to register each of your apps, then dump your epubs to DropBox and download from dropbox to each of the apps.
BookFusion can sync reading progress across Android, iOS and web application. I'm happily to use it to read my PDF books. Currently the reading experience is acceptable but not perfect. I've sent some feedback email, the support team says there will be a big update in Q3-Q4 2020.
I used to use Aldiko (and AldikoSync) on all my android devices, but this is now too much hassle as the device needs to be rooted to work (you used to be able to use an older version of Aldiko on un-rooted devices, but that stopped working on my latest android devices.) However this wouldn't work on desktops.
I now use Google Play books which, as "idiotprogrammer" states, works great across android (tablet and phones) and chrome on desktops. My next step is to find an e-reader that I can use Google Play books on (I have an old nook which I used for Aldiko, but it won't run GPB.)