My main pet peeve trope in all media is "immortality is a curse". I could understand it if they were old, crippled, tortured souls, but people who remain young, healthy and of sound mind complaining about not being able to bare life is peak copium on the writers' part. Some probably want to imagine it to be a curse just because they can't have it. It just doesn't make sense to me. Surely, just staying healthy and mentally agile is great! That's the whole point of all of medicine.
Well, yeah. None of us can be immortal of have super long lifespans, so we do cope with this by writing most immortal characters that way.
One notable exception in fiction (not necessarily manga/anime) that I've found is Belgarath from the Belgariad/Malloreon, by David Eddings. He was called the Eternal Man and was over seven thousand years old. An itinerate storyteller, powerful sorcerer, and a womanizing drunkard who seemed to enjoy life to its fullest (though he did have a mission from his literal god who existed on the same plane as him and talked to him directly, so that probably helped). He missed his wife terribly though, but that's only a natural response to her dying. There were a few other immortals around (including his daughter) that he kept company with on occasion.
In fact, the defining factor of whether or not your ageless immortal is miserable or not seems to be whether they've got company that lives as long as they do. Near ageless Elven societies in Tolkienesque fantasy stories that last for millenia mostly seem to be doing just fine, by comparison. It's just the poor shmucks who live alone, traveling from place to place and never forming real connections with people because it makes them feel sad when everyone around them is constantly dying who are
really miserable. I mean, hell, that sounds pretty bloody miserable to me too, and I
like the idea of sticking around for at least a hundred more years than I'm going to get, assuming it's not horrible, anyway. I wanna see what happens.
Bottom line, you want to make immortality appealing? Give your characters friends, family, and purpose. Hell, these things are what make life appealing in the first place.