You have to understand that there are not two indexes; one for desktop and one for mobile. There is just one.
If a page is found to be mobile friendly, it is marked within the index. If it is not, then it is marked with a reason code for Search Console to report. It is that simple.
Google hits your site just once per-page and assesses whether the page works for mobile. One thing people are not aware of is that the various Google bots are exactly the same except where they come from (what system places the request into the queue) and how they are tagged (for your information/understanding).
For example, testing a page for mobile friendliness is the same bot as googlebot and does exactly the same work. This is done to avoid crawling a page multiple times. What changes is what post fetch processes are required. There is a large overlap in the post fetch processes that allows Google to reduce the number of reasons to fetch a page again. This is done to be less obtrusive and save bandwidth- yours and theirs.
So yes, I expect that the desktop title tag should show depending on how the page was presented googlebot last. The reality is that the title tags should be the same mobile or not. This is something you need to fix. Either way, there is no differentiation from desktop or mobile in the SERPs except to say that a page is mobile friendly.