Redmine

One way to do it is to change *.yml file(e.g. en.yml file) - Make sure you change project to customer with "Whole Word only" option(i.e. in the string that's after : at every line). Remember to take the backup before making changes.

Do I have to reinitialise my redmine database after changing these?

No need to change anything in the database. The trick I mentioned basically displays something(like Customer) instead of Project. Please take a look at the answer given by Scott as below and compare en_orig.yml AND en_ticket.yml -
https://www.redmine.org/boards/2/topics/7953?r=7956

You can use similar tactic for project instead of issue.

It works, thank you for pointing out the way to do this.

Have a nice day.

Glad it worked!