If you care about having a modern, productivity-focused browser experience, Edge and Shift both come up short, but for different reasons.
Shift mainly exists as a wrapper for account switching and pinning apps on a sidebar. It feels like a thin layer on top of Chrome, not a real replacement, and hasn't kept up with newer browsers. Unless you specifically want quick account switching and sidebar app pins, Shift doesn't stand out. Most of what it does can be replicated with Chrome extensions like Workona or Toby, so it's hard to justify making the switch.
Edge, meanwhile, brings some AI features and tries to integrate tightly with Microsoft products, but its interface still feels dated and its unique features only really shine if you're deep in the Microsoft 365 ecosystem. If you're not, there's little reason to pick it, especially since its improvements mostly benefit Windows and Microsoft users.
If you're deciding between these two, Edge is the better pick only if you already live in Microsoft 365. Otherwise, neither is worth switching to for productivity alone, and you're better off sticking with your current browser and using extensions. Shift doesn't offer anything compelling enough to make it your main browser, while Edge at least offers some integration benefits for Microsoft-heavy users.