Fungi and animals have a common ancestor, but that ancestor was an unicellular protozoan, without any resemblance to a fungus, but which resembled somewhat a human sperm cell, by having a single posterior flagellum used for swimming. Fungi and animals and their close unicellular relatives are called opisthokonts (Opisthokonta), which is a name describing their posterior flagellum (opistho- means back, while konta was the name for the rods that were used for pushing boats in areas with shallow water).
The branch that has evolved towards animals has developed multi-cellularity while retaining their ancestral lifestyle of eating other living beings, which is a lifestyle that requires mobility.
The branch that has evolved towards fungi has adapted to a terrestrial life, unlike the branch that has evolved towards animals in the oceans.
To avoid desiccation on land, the fungi have developed a chitinous cell wall. This has solved the desiccation problem, but this wall has made the fungal cells immobile. So they had to change their lifestyle from the ancestral lifestyle of the eukaryotes to a lifestyle similar to that of the heterotrophic bacteria, i.e. fungi do not eat food by engulfing it, like animals, but they grow into food, by secreting enzymes that break the food into small molecules, which can be then absorbed by the fungal cells.
While there exists only a single group of living beings like the animals, which are both multicellular and mobile, there are several groups like the fungi, besides the true fungi. All such fungous organisms have immobile cells with cell walls, so if they are multicellular they must grow into food in the form of a branched network, in order to maximize the surface of contact between them and food.
The other groups of living beings that look like fungi, but which are not true fungi, are not closely related to animals. The most important of those groups is related to the brown algae, but there is even a group of bacteria that look like fungi, the actinomycetes. Besides the other living beings that feed like fungi, so they look like mycelia (branched networks), there are even more groups of living beings that do not feed like fungi, so most of the time they do not look like fungi, but which are terrestrial like fungi, so they must use the same method of spore dispersion by wind, so they grow mushroom-like bodies for the launch of spores, e.g. the slime molds.
This feels like the "we share 50% of our genes with bananas" factoid.
Considering humans have 3.1 billion base pairs and the most fungi have 30-300 million, I'm not sure how this could be true without some major caveats. Admittedly, a lot of our DNA is considered "junk" - though whether it all is or not is a question.
(the banana one is that they're talking specifically about protein coding genes which makes up about 2% of our DNA)
That's not inconsistent with GP's comment - animalia and fungi are opisthokonts, but siblings or cousins in a family tree sense, neither is ancestor to the other, but there is common ancestry.
