There are around 120 different groups and/or species of birds with “official” English common names that reference other birds (even more when including colloquial and/or rarely-used names), and around 50 more referencing non-bird animals. A few cases exist where two distantly-related bird taxa have the same two birds in their common names, but with the order reversed: cuckoo-hawks (Aviceda in part) and hawk-cuckoos (Hierococcyx), flycatcher-shrikes (Hemipus) and shrike-flycatchers (Megabyas and Bias - all three of these genera are fairly close relatives), lark sparrow (Chondestes grammacus) and sparrow-larks (Eremopterix), manakin-tyrant (Neopipo cinnamomea) and tyrant-manakins (Tyranneutes and Neopelma), and nightingale-thrushes (Catharus) and thrush nightingale (Luscinia luscinia). (There’s also thrush-like wren Campylorhynchus turdinus and wrenthrush Zeledonia coronata, and - when considering colloquial names - hawk-eagle (Spizaetus, Nisaetus, etc.) and eaglehawk, an old name for the wedge-tailed eagle Aquila audax). There are also a couple cases where two birds with the same “group” name are named after the same other bird, but with “-like” tacked on to one but not the other: lark bunting (Calamospiza melanocorys) and lark-like bunting (Emberiza impetuani), and shrike-tanagers (Lanio) and shrike-like tanager (Neothraupis fasciata).
Less common are birds with three different birds in their common name, of which I’m only aware of two examples: oriole cuckooshrike (aka eastern wattled cuckooshrike), Lobotos oriolinus (cuckooshrikes are named after a superficial resemblance to both cuckoos and shrikes, and this species is mostly yellow with a black head, like some orioles), and quailfinch indigobird, Vidua nigeriae (an obligate brood parasite of the quailfinch Ortygospiza atricollis, which itself is named after quail because of their vaguely similar plump, short-tailed shape).
Speaking of indigobirds, they’re specialist brood parasites, only laying their eggs in the nest of a single species or a small handful of species of estrildid finch (which they typically mimic in both vocalizations and nestling mouth patterns), and for two species of indigobird, the host species (or the presumed host species at the time of description) was referenced in the scientific name: barka indigobird, Vidua larvaticola (after the black-faced firefinch Lagonosticta larvata), and jambandu indigobird, Vidua raricola (after the black-bellied firefinch Lagonosticta rara). Unfortunately, it was later discovered that some of the supposed black-bellied firefinch song mimicry described and illustrated in the description of V. raricola was actually misidentified zebra waxbill (Amandava subflava/Sporaeginthus subflavus) song mimicry, and the type specimen of V. raricola was one of those zebra waxbill mimics (whereas the black-bellied firefinch mimics turned out to correspond to the previously-named V. camerunensis) - so V. raricola is stuck with a name that refers to a different indigobird’s host species!
