AutoCAD saves linetype in an external file, with .lin extension. You can import the linetype definition into a DWG file and use it in a drawing. However, it is not possible to export the linetype back to .lin file. At least not with an out of the box AutoCAD.
It means you can’t export a custom linetype from a file that you receive. The workaround is to copy an object from the drawing then paste it to your working drawing. Alternatively, you can use Design Center. It’s not always ideal, but it works well to copy styles from one file to another.
However, if you need to have the .lin file, you can use LINOUT from CAD Studio. You can download the utility here.
The utility is a Visual LISP Executables (.vlx) file. You can load it using APPLOAD command. After you load it, type LINOUT then enter. That’s it.
The .lin file is generated in the same folder with the same name as the drawing file. The file has all line types in the drawing, you may want to edit the file manually, remove what you don’t need.