Some sites have the same URL for both desktop and mobile content, but change their format according to User-agent. In other words, both mobile users and desktop users access the same URL (i.e. no redirects), but the content/format changes slightly according to the User-agent. In this case, the same URL will appear for both mobile search and desktop search, and desktop users can see a desktop version of the content while mobile users can see a mobile version of the content (2).
However, note that if you fail to configure your site correctly, your site could be considered to be cloaking, which can lead to your site disappearing from our search results. Cloaking refers to an attempt to boost search result rankings by serving different content to Googlebot than to regular users. This causes problems such as less relevant results (pages appear in search results even though their content is actually unrelated to what users see/want), so we take cloaking very seriously.
So what does "the page that the user sees" mean if you provide both versions with a URL? As I mentioned in the previous post, Google uses "Googlebot" for web search and "Googlebot-Mobile" for mobile search. To remain within our guidelines, you should serve the same content to Googlebot as a typical desktop user would see, and the same content to Googlebot-Mobile as you would to the browser on a typical mobile device. It's fine if the contents for Googlebot are different from those for Googlebot-Mobile.
One example of how you could be unintentionally detected as cloaking is if your site returns a message like "Please access from mobile phones" to desktop browsers, but then returns a full mobile version to both crawlers (so Googlebot receives the mobile version). In this case, the page which web search users see (e.g. "Please access from mobile phones") is different from the page which Googlebot crawls (e.g. "Welcome to my site"). Again, we detect cloaking because we want to serve users the same relevant content that Googlebot or Googlebot-Mobile crawled.