The easiest way to rename your workspaces is to install the (official) gnome extension Workspace Indicator. Through its configuration settings (you can access these in Gnome Tweaks), you can provide custom names for the workspaces. Just turn the extension off when done renaming if you do not want the workspace indicator in your panel.
In the body of your post, you mention you may want to get rid of the names altogether. That effect can be achieved by setting the name to a single space in Workspace Indicator.
As a bonus tip, perhaps you like an entire clean approach where no popup at all appears when you switch workspaces using the keyboard. To achieve that, install the Disable workspace switcher popup extension.