Welcome to Atlassian Community!
In order to create a classic project (or company-managed as it is being renamed to) can only be created by users that have Jira admin access, so you would have to add the user to your admin group for Jira in order to allow them to create the project.
Natively, Classic Projects can only be created by Jira Admins.
This is because - unlike with Next-Gen projects - Classic Projects can have shared configuration such as workflows, screens, permissions, etc, that also require Jira Admin to update.
There are some options using apps - see Delegated Project Creator as one option.
adding to the perfect clear and correct answers of Ste and Mikael I just wanted to link an article which I found to be helpful. This one might answer some other upcoming questions in regards to the two types of projects alongside with their respective differences:
Atlassian's marketplace partners have had a very productive start to 2021! Since our last roundup, our developer community has added over 160 new cloud apps to the Atlassian Marketplace to help you...
Connect with like-minded Atlassian users at free events near you!Find an event
Connect with like-minded Atlassian users at free events near you!
Unfortunately there are no Community Events near you at the moment.Host an event
You're one step closer to meeting fellow Atlassian users at your local event. Learn more about Community Events