Minneapolis and Austin are very much alike in many ways, but you're right. I think The sensibilities of chefs/restaurateurs are very similar between the two cities. For the most part it's young, classically trained chefs that don't give a shit about exclusivity, status, or Michelin stars. They just want a little 5 to 20 table spot, whether it's in a warehouse district or the middle of a neighborhood, to cook the food they love to eat. They care most about the ingredients, customers, and staff (in that order).
There is definitely nothing approaching Alinea here (no one is trying and it wouldn't work anyway). If you want a fancy, romantic establishment your best bet is the Lexington.
Aside from that, there are a few trends that are hot right now here: fancy pizza, southern comfort, smoked meats (at 3x's the price of better food down south). That's fine for us locals but I can't recommend any of it to a traveler- well maybe some fried chicken and greens at Revival if you're here for a while.
The true culinary identity around here is "farm to table" with ingredients that are locally sourced, and technique that is decidedly French (with a smidge of Vietnamese). Expect small menus and small plates meant for sharing.
If I were pushed right now to recommend 3 "fine dining" dinners to an out-of-town friend, I think it would be Bar La Grassa, Spoon and Stable, and Travail Kitchen Amusements. You can't go wrong with any of the other ones I listed though.