Ah, sorry about that, since your link on your desktop site has a parameter, I assumed that you were setting up a separate mobile site (which is what you referred to, with a different subdomain) rather than a dynamically served site, which is what you actually have.
Which means you're absolutely right, you don't need links between the two pages.
In that case, I think you're okay. I just double checked on Amazon.com, which is probably the most famous, most powerful dynamically served site in the world. Its mobile version of the homepage has the code:
<title>Amazon.com</title>
But, mobile search results for "amazon.com" return:
Amazon.com: Online Shopping for Electronics, Apparel, Computers ...
Online retailer of books, mobies, music and games along with electronics, toys, apparel, sports, tools, groceries, and...
That's the page title and meta description of the desktop version of the site.
I'll admit, I've never seen Google admit to using the desktop version of the page title and meta description before. But, if Google's doing it to Amazon - whose site we know Google crawls and understands - I think you're okay. Like you said, you have the "mobile friendly" designation, so my guess is, this is a Google decision, not a mistake on your part.
I hope that eases your worries a bit! Really, I don't think there's anything you need to do here. And, I really appreciate you bringing this into the community. I learned something today!
Best,
Kristina