His pride is literally all he has left. When they were kids, neither of them had money, power, or influence. Now Taemin has, well, nothing. On top of all his own problems, he lost his own body! Now Woojin is stronger than him and more confident and just generally has the upper hand in life. Admitting he's Taemin isn't about confirming the truth, Woojin already knows he's Taemin, and Taemin knows that. It's about being vulnerable and acknowledging their past and letting go of his pride. And Taemin is also so bitter and angry and hurt and betrayed that Woojin left him when they were kids. And the guy Woojin became isn't going to make it easy for him.
His pride is literally all he has left. When they were kids, neither of them had money, power, or influence. Now Taemin has, well, nothing. On top of all his own problems, he lost his own body! Now Woojin is stronger than him and more confident and just generally has the upper hand in life. Admitting he's Taemin isn't about confirming the truth, Woojin already knows he's Taemin, and Taemin knows that. It's about being vulnerable and acknowledging their past and letting go of his pride. And Taemin is also so bitter and angry and hurt and betrayed that Woojin left him when they were kids. And the guy Woojin became isn't going to make it easy for him.