When classifying living beings, a single classification criterion is not good enough.
One classification criterion is descendance from a common ancestor, i.e. cladistic classification.
In many cases this is the most useful classification criterion, because the living beings grouped in a class defined by having a common ancestor share a lot of characteristics inherited from their common ancestor, so when using a name that is applied to that class of living beings, the name provides a lot of information about any member.
However there are at least 2 reasons which complicate such a cladistic classification.
One is that the graph of the evolution of living beings is not strictly a tree, because there are hybridization events that merge branches.
Sometimes the branches that are merged are closely related, e.g. between different species of felids, so they do not change the overall aspect of the tree. However there are also merges between extremely distant branches, like the symbiosis event between some blue-green alga (Cyanobacteria) and some unicellular eukaryote, which has created the ancestors of all eukaryotes that are oxygenic phototrophs, including the green plants.
Moreover, there have been additional symbiosis events that have merged additional eukaryote branches and which have created the ancestors of other eukaryote phototrophs, e.g. the ancestor of brown algae.
After any such hybridization event, there is the question how you should classify the descendants of the hybrid ancestor, as belonging to one branch or to the other branch that have been merged.
For some purposes it is more useful to classify all eukaryote phototrophs based on the branch that has provided the main nucleus of the hybrid cell, and this is the most frequently used classification.
For other purposes it is more useful to group together all the living beings that are oxygenic phototrophs, including various kinds of eukaryotes and also the blue-green algae, and divide them based on the evolution tree of their light-capturing organelles, i.e. the chloroplasts.
This is also a valid cladistic classification, because all oxygenic phototrophs, both eukaryotes and prokaryotes, are the descendants of a single common ancestor, some ancient phototrophic bacteria that has switched from oxidizing manganese using light energy, to oxidizing water, which releases free dioxygen.
Even when there are no branch merges due to hybridization, there remains the problem that in the set of descendants from a single ancestor there are some that are conservative, so they still resemble a lot with their ancestor, and some that are progressive, which may have changed a lot, so they no longer resemble with their ancestor.
In this case, using the name of the entire group provides very little information, because most characteristics that were valid for the ancestor may be completely inapplicable to the subgroups that have become different. In such a case, defining and using a name for the paraphiletic set of subgroups that remains after excluding the subgroups that have evolved divergently may be more useful in practice than using only names based on a cladistic classification. For instance the use of the word "fish" with its traditional paraphiletic meaning, i.e. "vertebrate that is not a tetrapod", is very useful and including tetrapods in "fishes" is stupid, because that would make "fish" and "vertebrate" synonymous and it would require the frequent use of the expression "fishes that are not tetrapods", whenever something is said that is correct only for vertebrates that are not tetrapods, or of the expression "bony fishes that are not tetrapods", for things valid for bony fishes, but not for tetrapods.
While in many contexts it is very useful to know that both fungi and animals are opisthokonts, and there are a few facts that apply to all opisthokonts, regardless whether they are fungi, animals or other opisthokonts more closely related to fungi or more closely related to animals, the number of cases when it is much more important to distinguish fungi from animals is much greater than the number of cases when their common ancestry is relevant.
Animals are multicellular eukaryotes that have retained the primitive lifestyle of the eukaryotes, i.e. feeding by ingesting other living beings, which is made possible by cell motility.
Fungi are multicellular eukaryotes that have abandoned the primitive lifestyle of the eukaryotes, and which have reverted to a lifestyle similar to that of heterotrophic bacteria, just with a different topology of the interface between cells and environment (i.e. with a branched multicellular mycelium instead of multiple small separate cells).
This change in lifestyle has been caused by the transition to a terrestrial life, which has been accomplished with a thick cell wall (of chitin) for avoiding dehydration, which has suppressed cell motility, making impossible the ingestion of other living beings, the same as for bacteria. Moreover the transition to a bacterial lifestyle has also been enabled by several lateral gene transfers from some bacteria, which have provided some additional metabolic pathways that enable fungi to survive when feeding with simpler substances than required by most eukaryotes, including animals.
So even from a cladistic point of view, fungi have some additional bacterial ancestors for their DNA, besides the common opisthokont ancestor that they share with the animals.
Animals are unique among eukaryotes, because all other multicellular eukaryotes have abandoned the primitive lifestyle of eukaryotes, by taking the lifestyles of either heterotrophic or phototrophic bacteria. However for both other kinds of lifestyle changes there are multiple examples, i.e. besides true fungi that are opisthokonts there are several other groups of fungous eukaryotes that are not opisthokonts, the best known being the Oomycetes. There are also bacteria with fungal lifestyle and topology, e.g. actinomycetes a.k.a. Actinobacteria.
If we will ever explore other planets with life, those living beings will not have a common ancestor with the living beings from our planet, but nevertheless it will still be possible to classify them based on their lifestyle in about a half of dozen groups that would be analogous to animals (multicellular living beings that feed by ingestion, so they must be mobile or they must have at least some mobile parts), fungi (multicellular beings that grow into their food, absorbing it after external digestion), oxygenic phototrophs, anoxygenic phototrophs, chemoautotrophs, unicellular equivalents of animals and fungi, like protozoa and heterotrophic bacteria, viruses.
These differences in lifestyles are more important in most contexts than the descendance from a common ancestor.
So while it is useful to have the name Opisthokonta for the contexts where fungi and animals and their close relatives must be included, it is much more frequent to need to speak separately about fungi and other fungous organisms on one hand, and animals on the other hand.
I agree that the term "kingdom" is obsolete when used in the context of a cladistic classification of the living beings.
Perhaps it should be retained for a non-cladistic classification of the living beings, based on the few fundamental lifestyles that are possible, and which would remain valid even for extraterrestrial living beings.