Here's why:
* Assign means to give someone a specific task or responsibility.
* Assignment refers to the task or responsibility itself.
Therefore, you would assign an assignment to someone, meaning you would give them a specific task or responsibility to complete.
For example:
* "The teacher assigned an essay assignment to the students."
* "The manager assigned a new project assignment to the team."
You wouldn't say "make an assignment" because "make" implies creating something new, whereas "assign" implies giving something pre-existing.