The Assignee field is a single-value one by design. You can't make it store multiple users.
What you could do is adding a "User picker" type custom field to your issues and allow multi-selection there. You could even call your custom field Assignees and hide the original Assignee field.
Don't forget though that the built-in Assignee has a special use in many parts of Jira! For example, it is shown on cards. It is used by the "Assigned to me" gadget. And so on. You will lose this special features, but it doesn't matter, then it could be a good solution to you.
Even with these, I think that the better approach would be splitting the work to sub-tasks and assign the sub-tasks. For example, if you wanted issue X be assigned to Alice and Bob, then add two sub-tasks that are clearly defined, and assign one of them to Alice and the other to Bob. It is easier to understand, matches Jira conventions better and can be tracked more precisely.
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