Well, we are as much alike as our consciousness/brain can manage to find a pattern for. The consciousness compresses reality by finding patterns. As long as we are able to understand some phenomenon, there will always be a pattern that this understanding is serviced by, and we will be able to see some relation between two different things through that pattern. In that sense, everything we perceive will be connected in one way or another. In that sense, everything is "one".
But does that say anything about the physical world, or rather about our own brains, our consciousness and how it works? A matter of perspective. And of how you define "real".
Fungi and animals actually have a common ancestor, protista. It's not exactly my area of expertise, but I believe that the way a fungus grows its tendrils and communicates over it is much like our nervous system.
Here[0] is a fancy graphic that succinctly displays how the kingdoms of life are categorized.
Protista is a paraphyletic group of single-celled eukariotes. However, animals and fungi are each other's closest multi-cellular living relatives. Animals are closer to other holozoa, and fungi are closer to other holomycota, but both are equally distant from, for example, Amoebozoa, which includes many other protists.
I attended a course of neurophysiology, and the most stunning thing I've found was that all (or maybe almost all, I don't remember clearly) the neurotransmitters used by a human's neurons exist in a single cellular organisms, even in prokaryotes. Serotonin, glutamat, dopamin and others could be found in bacteria, so bacteria had all the chemistry needed for a central neural system hundreds of millions years ago, and nevertheless it took hunderds of millions years ago to build Einstein's brains.
But does that say anything about the physical world, or rather about our own brains, our consciousness and how it works? A matter of perspective. And of how you define "real".