The negative development of Solid Snake's sense of identity is one of the biggest tragedies in this series.
CLARIFICATION; Negative development does not mean bad, it just means character development that goes in the opposite direction.
From the beginning, we see Snake, or David, fight the idea he's just a manufactured clone. That who he is as a man still matters. He considers his real name David, and sees himself as human. He had huskies he cared for and raced with, it wasn't much, but the game ends with him cheerful towards the future with Meryl or Otacon.
By Sons of Liberty, Snake has accepted he's a clone, but still sees himself as a person. One who can still pass on his experiences down to the next generation, even if he's not passing down genes. His origin doesn't matter to him, he's sure of himself and his identity, and knows he'll still be able to pass down part of himself to the future.
But by Guns of the Patriots, the rapid-aging revelation broke him. He's still a soldier who will finish the mission no matter what, but he no longer sees himself as a man. He's fully accepted Liquid's reality - they are merely shadows, freaks born out of a test tube. He doesn't see himself as "David" anymore, or someone he'd even consider human. Even after Big Boss' pep talk, his remaining days he acknowledges will be living as a monster who won't pass down anything, with the only solace being Otacon promising to pass down his story.
The "Beauty and the Beast" analogy is his mindset. He'll never be human, he'll never restore his youth, he'll simply wither away decades before his time simply because he was CREATED to.
It's so depressing, man. He deserved better.