You can move it into another geodatabase and ArcGIS should also copy the associated domains with it. Like you discovered, Shapefiles do not support field domains; it needs to be in a geodatabase (File or Personal). And, as Leo D. mentioned, a feature class can only exist in a geodatabase.
If you need to deliver the data to someone in a format other than a geodatabase, you have two obvious options:
Then I create a Feature Class with a name and alias, select "Point Features" and choose the Coord system and click Next until I get to the Field Name and Data Type table. There, I associate the feature class to the domain as described above and select Finish.
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