I am implementing AMP for landing pages on my website. From a related question on StackExchange, I learned that the latest 2 versions of major browsers support AMP even on desktops. In that case, if I use the same page both for desktop and mobile and a user happens to access the page from an unsupported browser, will AMP still render properly? Is there a mechanism for fallback to normal HTML?
AMP is just HTML/CSS/JS, that's why it works in all modern browsers. Browsers don't really "read AMP", they read HTML/CSS/JS because that's what AMP is comprised of. From the AMP documentation:
AMP HTML is a subset of HTML for authoring content pages such as news articles in a way that guarantees certain baseline performance characteristics.
[...] it does not require the development of new rendering engines: existing user agents can render AMP HTML just like all other HTML.
Also, AMP HTML documents can be uploaded to a web server and served just like any other HTML document; no special configuration for the server is necessary.
Regarding browser support, according to Google:
the core AMP library and built-in elements should aim for very wide browser support and we accept fixes for all browsers with market share greater than 1 percent
So while AMP may not render perfectly in all browsers, neither will standard HTML. Unless you are targeting IE < 10 support, your AMP pages should render well enough even on somewhat out-of-date browsers.
All that said, it makes the most sense to directly test your AMP pages in the specific deprecated web browsers that you know your visitors are using - as there's no sense in supporting browsers that no one is using!
AMP is intended to be a copy of your original page. You should always have a regular HTML version of every page, and consider that the primary copy, the one that most visitors will see - no matter what size screen they're viewing your site on.
The AMP version is only intended to be displayed when the visitor is using a mobile device and coming straight from Google search results. Because Google is the one showing the result, it then uses one of two technologies - either Google AMP Viewer or a Signed Exchange - to display the page. So, as long as visitors are coming from organic Google search using a mobile device, even if they're using an outdated browser, Google provides an intermediary to parse and display the content in the mobile browser. But visitors who come to your site any other way - searching Bing, clicking on an ad that links to your site, or from search results on a larger screen - should see the regular HTML version of the page.