What is the relationship between identity and personality? I would like to know whether identity and personality are the same thing, as they both answer the question of who you are. 
Nevertheless in a cognitive approach, identity is considered a part of personality, concretely it's considered as a brain structure inside of the memory.
I think the relation is more or less subjective. 



Identity is something that you give yourself. It has to do with what you stand for, morals, values, etc. 

It is who your are physically and legally, but that's just a basic "you already know that" statement.

Personality is the way in which you portray or "live in" your identity.



For example, you can identify parts of someone's personality: humorous, attractive, intelligent, funny. 

Both adapt and change over time, but your identity changes less often, I believe. Together, they are what makes a whole person.

It's really difficult to give a concrete answer because a lot of this can be subjective, debatable and complex.



It's also important to remember that while in some cases your personality and identity may be the same, in the end, they serve different "roles." If someone says "you're unique", it's true. Both your identity and personality are what makes you unique. They help each other in that way.

Let's take a look at this from a different angle: Let's say that we know this guy named Tom is a murderer. That is his identity. His personality, however, may not portray that and can come off as fun or charming. Personality is how someone behaves. They both go really well together but that doesn't mean that they are always similar. In most cases, you will find that they are similar, though.



Sources


Personality vs. Identity | Authentic Systems - Motivation Research &amp; Development
Discover Your Personality | Truity
The Difference Between Personality &amp; Identity | HiNative