Just like Voldemort came up with his own name to disown the muggle Riddle family last name, Snape’s self-title as the half-blood prince does the same. This is also a way to mislead the reader into believing that Snape is truly on Voldemort’s side, which seems apparent at the end of Half-Blood Prince. However, as we learn in The Deathly Hallows, Snape has an even deeper underlying motivation.
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Source: https://www.distractify.com/p/why-is-snape-the-half-blood-prince