Do all mushrooms have the same multicellular ancestor? Did that ancestor feature common mushroom elements such as cap, stem, sporofore, vulva, bulb, etc.
All fungal species within the kingdom have the same common ancestor, which is unicellular (thought to be protistan). This common ancestor is the point at which it is thought that animals diverged, as suggested by this article. In their words:
A fungal phylogenetic tree is shown below, which originated from this paper.
Note: all colored areas represent different phyla of fungi.
I will respond to a few questions addressed in the comments.
I am rather sure (but can't explicitly confirm with a reference) that multicellularity in fungi only evolved once. If this were true, I would also hypothesize that mushroom-like characteristics also evolved only once, thus indicating that there is a multicellular ancestor to all mushrooms.
No. One branch of dikarya seems to consist of only unicellular organisms, suggesting that multicellularity evolved at some point within organism under only one dikaryan branch.
No. See previous answer.
I am not exactly aware of the similarities in fruiting bodies in both branches, but if the physiology behind fruiting bodies are similar, it is very likely because the dikaryan common ancestor had a similar mechanism.