I ramble a lot here so there’s a tl;dr at the bottom.
The Horde’s current identity crisis stems from the Garrosh era.
The WC1-WC2 era Horde has a very clear identity–demonic army bent on conquest and slaughter. Cut and dry. Forest Trolls joined for a chance to reconquer their lands, and get even with old enemies. Goblins joined because money (?).
The WC3 Horde is a union born of mutual defense from hostile native elements in their new homeland. The orcs recruited the dark spear trolls during their exodus to Kalimdor, and the trolls settled in Kalimdor along with the orcs. The orcs assisted the Tauren in the Barrens against the centaur, and in return the Tauren provided troops and supplies to the orcs as they traveled to Stonetalon.
Both the Trolls and Tauren joined the Horde in an official military and defensive union during Admiral Proudmoore’s invasion of Durotar (the plot of the ‘Founding of Durotar’ campaign from WC3). This union persisted following the invasions end likely to ensure mutual military support in case of future possible problems (worth noting Thrall had excellent diplomatic relations with both Cairne and Vol’jin). THIS would be the “family” Horde that’s been previously referred to, and these three races have always felt very close (just as an example Crossroads in Classic has Tauren and Orc guards which shows the military cooperation, Trolls had their own section in Orgrimmar, small things like this). It’s a union born of mutual trust and spilled blood. It feels very natural.
So now we have the Classic era Horde. The big change is the addition of the Forsaken, an addition that has always been somewhat contrived and you feel it in their quests. There’s a clear divide between WC3 Horde Quests, and Forsaken “Horde” quests. As such the Classic Horde essentially feels like the WC3 Horde, but with these new associates who are radically different and feel totally apart from the original three races (which canonically they were). If they were a family member, at best they’d be an adopted sibling you don’t really like or consider a sibling.
The TBC Horde adds Blood Elves, also a somewhat contrived addition. The BElves also are very much like the Forsaken in that they also feel more like another set of new associates who are radically different and totally apart. This iteration is more or less the same in Wrath.
Garrosh’s Horde changes this radically, and part of the problem is the side lining of the other races almost completely in favor of more and more orcs in the spotlight. From here you really only start to see Orc soldiers, quest givers, and architecture (referring to the Cataclysm remakes + new zones, which full disclosure I haven’t played through in probably six+ years now, so I’m going off the impressions in my memory). So instead the Horde feels like this monolithic orcish entity, and hey sometimes there’s these other guys here.
tl;dr The Horde has never had a family style identity beyond the original three WC3 races (Orc-Tauren-Troll). Each addition afterwards has been somewhat contrived and those additions have always stood apart and had their own distinct experience typically apart from the original three. The identity crisis is because the Horde never bothered to properly integrate the additional races (though from screenshots and gameplay I’ve seen they do at least try now by having stuff like BElf soldiers alongside Orc ones, etc).