My only experience with mushroom compost is from my non-cannabis landscaping days, 25 or so years ago. Don't take this as gospel, but my boss told me that it was sheep shit that had been used to grow mushrooms on. (Google says it can also be other things.) Apparently, after it's too spent to produce mushrooms at commercially viable rates, it gets sold as a garden amendment. I think it's supposed to be pretty good stuff. Since it's already had mushrooms grown on it, I think you can rest assured that it's sufficiently composted to grow plants in. We used it a lot when planting shrubs and trees; we'd dig a nice big hole and then mix the compost 50/50 with the native soil to refill the hole, then plant into that. We'd also rototill it into beds that needed some organic matter, usually adding about 4 to 6 inches. Everything we planted always did fine.
I just found a place that sells it in bulk for about the same price that I've been paying for the "green bin" compost with the plastic bits in it, so I'm going to buy a yard and have a good look at it here pretty soon. If it doesn't have any crap (garbage, I mean) in it, I'll probably buy a whole bunch.
In terms of how it stacks up against animal manures, I would say that mushroom compost is compost while composted manure is fertilizer (used like guano).
Mushroom compost can be hot.... I lived in the mushroom capitol of the world for several years and used mushroom compost a lot in our gardens. It comes from horse manure and it is manure. Not sure what that stuff is you buy in the bags... we used to go (and still do) and pick up a load.. or have a few yards delivered. There's a huge mushroom industry here in SE PA. We don't live in Kennett Sq. anymore but still get the mushroom compost.
So here's the skinny.. you can use this stuff but be aware it "can" be hot. What they do with mushroom soil is mix horse manure with straw and a few other amendments and sterilize it, conveyor it into the mushroom houses and inoculate it with mushroom spores, turn off the lights and turn on the fans. So its still fresh manure when they use it to grow in. So buyer beware, it can be
not composted enough and burn your plants roots.
Mushroom "soil" I think is the stuff that is composted. Read the label if in a bag. If it's in a big pile at the nursery, ask how long it's been composting (sitting). Should be at least 2 years old...and not hot. You will be able to tell if it's hot, it will be warm to the touch and steamy when opening the bag or digging into the heap.
Manure composting is more than shoveling manure into a pile... believe it or not. Mushroom compost is just that... they pile up the horse manure and there you have it. Basically its NOT composted or not composted properly for use as a soil.
This stuff is good for outdoor gardens. I don't use it in container gardening... there are better options "for me". Home made is the best followed by Coast of Main lobster compost or Malibu's BU Blend compost. It's an excellent mulch, as it's been sterilized and HOT so nothing will grow (weeds) in it for several years. So that right there (good use as a mulch) should be your caution for use in containers for soil building.
Permaculture ... ahhhh while reading about it, also check out Biodynamics and Rudolph Steiner. Both ideas are steps toward "sustainable living", and of course farming is part of that. Permaculture is a bit less structured... in that it is "generally" organic, where Biodynamics is sustainable organic living.
I got the use of Horse Tail Fern tea to combat mold from Biodynamics ... There are other great things we can take from both. The idea we can live sustainably is certainly a good thing.