I too have observed that if safari or an app with a wkwebview is split paned with an another app and the first app's width is compact, the displayed website will load the mobile version of the site. Ditto for bringing up Safari or a WKWebView app in a compact width popup. Resizing the safari or app w/WKWebView to a regular width causes the desktop version to be loaded.
I think this is a feature, but am not 100% certain.
Are there any news on this (maybe with the latest version of iOS 13.3?) - not being able to set the userAgent makes it difficult to request the mobile version of a web page.