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.
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:
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