No. Find any other partner device.
Microwave + garbage disposal, fine. Microwave+dishwasher, fine. Microwave + 1 of 2 kitchen receptacle circuits, fine.
Nothing shares a circuit with a refrigerator.
Because food spoilage is serious business. You really don't want a situation where someone comes into the kitchen, looks at the time, the microwave is dead, so they go downstairs and reset the breaker then set the microwave time. Hours later they grab some food out of the fridge, it's nice and cold, so they eat it. Unbeknownst to them, the fridge has been off for awhile, the food spoiled, and they just poisoned themselves.
A separate refrigerator circuit isn't a Code mandate, but it it really improves safety for less than the cost of a GFCI outlet.
NEC does not require kitchen refrigerators to be on any protective circuit - not GFCI nor AFCI. That's what you want. You don't want dueling safety systems, and frankly, a refrigerator is not a shock risk warranting a GFCI.
Of course you know, the two breakers must be handle-tied. The handle ties must be UL-listed and made for the breaker (i.e. By the manufacturer). Those are pricey and hard to chase down, so we recommend a 2-pole breaker instead. It doesn't cost anymore than two 1-poles, and they're easy to find. Handle-ties don't guarantee common trip, but they usually cause it... And 2-poles do guarantee it!
MWBCs are obsolete, anyway
The larger problem with MWBCs in general is they are being made obsolete by the new requirements coming down the 'pike, like AFCI and GFCI on increasing numbers of circuits. And those absolutely require 2-pole breakers, you cannot use 2 singles and a handle-tie (expept GE AFCIs, and those don't allow GFCI at all).
There's also a problem when people use "double-stuff" duplex breakers. A lot of people cheerfully throw a MWBC on a duplex, but they are overloading the neutral because both sides of a duplex are on the same pole of power. Also they can't be handle-tied, so there's a codevio right there. (You can't improvise a handle tie with a nail, you need a listed tie, and no manufacturer sells that).
I have basically stopped installing MWBCs, although I will do them if I expect interest in 240V appliances on that circuit. An MWBC allows you to tap hot and hot for a 240V load. So if you have a chef with European appliances, or 240V tools in your shop, MWBCs are a great option then - but you will also be using a 2-pole GFCI+AFCI breaker, which means you won't be in a GE panel.