Well, here's the thing. Khada Jhin is an alias because Jhin is smart enough to not go around killing under his real name. Having that identity as Khada Jhin safeguards him from becoming a pariah outright from his gruesome art, even though he was eventually exposed by the higher-ups of the Kinkou order.
Meanwhile, Tahm Kench and the Kindred are mythological figures, and since they're present in multiple religions, they are bound to have many names and many interpretations on their being. Tahm Kench, in Bilgewater, is known as the River King, and Tahm Kench is the name of some gambler he's devoured and taken the name of - so the story goes. The Kindred, The Lamb and Wolf, Ina and Anu, whatever you call them, they are finality. It's like how we personify Death in multiple ways ourselves - the Pale Rider, the Grim Reaper, Death, the Ferryman, whatever. It's a mythical figure, a personification of a concept.
As for Bard - he's a little more difficult a subject to touch over. Bard is, unlike Tahm and Kindred, a real-world force. He's a Celestial, like Soraka, a being of the Stars themselves. Bard has a rarely evoked but nevertheless MASSIVE impact on Runeterra. And because of how he saved the balance of the universe with people even viewing him, he's been woven into story. He's attributed to a star-sign, and while nobody knows his true name (perhaps except for Soraka), those who do know him know him as his star-sign. Bard. And it's like what we do with our own zodiac - Aries through Pisces, Ophiuchus, Cygnus, Crux, etc., or how the Romans attributed their Gods and Goddesses to celestial bodies in our solar system - Pluto, Mars, Venus, and Neptune, are good examples. We know enough about Bard to say with certainty he exists, but we've never met him in-person or been given the opportunity to understand him, so we simply call him by what we know him as - a musician of great influence. A Bard.