Pinning tabs is possible in normal window but not in private browsing. We can a reverse problem before where pinned tabs in normal window would appear in private browsing (https://orionfeedback.org/d/3496-pinning-tab-in-normal-window-shows-the-freshly-pinned-tab-in-currently-open-private-windows-too-including-session) and probably to solve that we have removed ability to pin completely.
Safari allows pinning in private browsing. Firefox does too. Firefox will delete all pinned tabs when the private browsing window is closed, which is probably the behavior you'd expect, otherwise in database you'd have favicons and links again being retained - Please don't change that. Current setup is sweet and literally leaves no trace of any browsing at all in any folders when in private browsing and asked Orion to 'disable history'.
But temporary pinning of tabs while window is open should be a normal behaviour instead of just disabling it completely.