Memory and identity are inseparable. Our accumulated experiences are how we form a conception of what it means to be us. If you suddenly forgot everything you've done, you won't know who you are. What if you start thinking you're a high-school student when you're really a rocket scientist turned astronaut, or fifty-three aliens simulating a human life-form? Hey, it's unlikely, but you would never know.
In Eternal Sunshine of the Spotless Mind, Clem and Joel both have their memories wiped so that they can forget each other. Their memories of the last two years centered on each other, and now, with their memories wiped, they feel lost and empty. It's not a surprise, because they have both lost pieces of themselves in the erasure process.
Questions About Identity
Is Joel's identity affected when he re-experiences his memories? Does he become, however briefly, a different person by gaining insight into the person he is or was?
Is there evidence that Joel's and Clem's identities have rubbed off on one another? Is Joel in any way more spontaneous or Clem more reserved?
Do Clem and Joel have an accurate idea of who the other is? Is their struggle a misunderstanding, or is it deeper rooted than that?
Chew on This
You can never really change who someone is. Even though Clem and Joel have had some memories erased, they're still the same people they used to be, minus a few kisses and arguments.
Joel and Clem can never have the relationship they first had, because their identities have been altered by the information that they were once together. This will always be with them, affecting the decisions they make and ultimately affecting who they are as individuals.