If it were the case that two "sperm" or two "eggs" can create a zygote too, then the "differences" between a "sperm" and "egg" would be arbitrary and could exist between two "sperm" or two "eggs", because two "sperm" or two "egg" could do the exact same thing a "sperm" and "egg" do.

Logic fail. There are often myriad ways to achieve the same end, but that doesn't mean the different methods in each case suddenly become the same. If I want to go to X place, I can drive myself in a car or truck, take a train, bus or plane, hire a car & driver, hitchhike, ride a motorcycle or horse, or take a horse-drawn carriage or cart. If I have the time, I also could ride a bike, unicycle, scooter, wheelchair, tractor or golf cart there; take a hot air balloon; or run, jog, walk, rollerblade, skip, hop &/or pogo-stick my way there. If the place I am going is on the seacoast or a river & my place of departure is adjacent to the same body of water, I could swim, sail take a motor boat, water taxi, jet ski or paddle board there. I could also parasail.

The fact that I'd land in the same destination at the end doesn't mean that the differences between all these varied modes of transport become "arbitrary" and are elided. It also would would NOT mean there is no travel by land, air or water by a range of diverse means - & that instead, it's all just simply travel.

Some organisms have evolved to use both asexual and sexual means of reproduction. For example aphids, slime molds, sea anemones, and some starfish are capable of both asexual and sexual reproduction. This doesn't mean that there is no difference between the sperm & eggs of those species. And none of what these other animal species do has anything to do with humans & other mammals.