There is no "copying and pasting a line type ", but maybe copying line styles, weights or patterns between projects. Anyways this won't affect the project were you are pasting the line to, because the line style from the host project is overriding the line style you are trying to copy. This makes sense so you won't override line styles, weights or patterns just by copying.
If you want to transfer the same line style from one project to another use transfer project standards:
- open the 2 projects (origin and destination)
- Manage > Transfer project standards > (select) Line styles
- Overwrite the line style
If you do this, the line weight, color AND pattern associated to the style will be overwritten.