Partly. Amazon makes no Kindle app for Linux, but read.amazon.com opens your Kindle library in a browser once you sign in, and your Kindle device and phone app carry on as normal.
Kindle by Amazon · Notes & PDFs · Works in a browser
Amazon's reading apps page covers iOS, Android, Mac and PC. There is no Linux version and no sign
of one.
The route that works is the browser. Go to read.amazon.com, sign in with your Amazon account, and
your Kindle library opens for reading, with your place, bookmarks and notes synced the way they
are on your phone or Kindle device. For most people that is the whole answer.
Your books are not lost, but they are not portable either. Kindle books carry copy protection.
calibre, the ebook library app everyone will point you at, says plainly that this protection stops
it opening those files, so you cannot simply drag your Kindle purchases into calibre or Foliate
and read them there. That is an Amazon decision, not a Linux one, and it applies on Windows too.
What does work on Linux is everything not bought from Amazon. EPUB files from an author, a public
library service, Project Gutenberg or a bookshop that sells them unprotected open straight away in
Foliate, and calibre will organise them and copy them onto a Kindle device over the cable.
A pleasant reader for books you hold as ordinary files. It will not open Kindle store purchases, so use it alongside the Kindle site rather than instead of it.
Your files: Opens EPUB, comic archives and PDFs. Protected Kindle books will not load.
Keeps an ebook collection in order, converts formats and copies books onto a plugged-in Kindle. Your existing calibre library folder moves across as is.
Your files: Manages unprotected ebooks and can copy files onto a Kindle device. It cannot open protected Kindle store purchases.