Is it possible in Forge to store multiple named values within a single custom field and still have Jira track the change history of each value individually?

Well, yes and no. In Jira Cloud, the audit log tracks that a change was made to a field, however, it doesn't record what individual parts of the field's content or values were changed. You could get the change log of the Issue to know what the entire field's content or value was before and then after the change, but you'd have to parse that and compared the two to determine what specific 'individual' part within the field changed... and that's adding more complexity than just using multiple custom fields.

Rather than use custom fields to store information for the Issue, would entity properties work in your use case? They are flexible, searchable and unlimited per Issue.