I'm pretty sure the 'Half-Blood Prince' thing is supposed to sound racist in-universe, as that's someone of mixed heritage (magic and non-magic) referring to hi

I'm pretty sure the 'Half-Blood Prince' thing is supposed to sound racist in-universe, as that's someone of mixed heritage (magic and non-magic) referring to himself self-deprecatingly in a universe where racism against non-magical heritage is very much a real thing and a term like 'mudblood' is considered an extremely offensive racial slur.