It does, actually.
You’ll need to clarify more specifically what you mean by “task information” but my guess is you have custom field values set on a task in one project and when you move that task to another project, you’re not seeing those custom field values.
If that’s what’s happening, it’s because you don’t have those custom fields set to be included on the second project. If you make sure the custom fields in question are in your organizational custom fields library, and then you add them to the second project from the library, then you’ll see values on tasks for those custom fields.