What is it called when you get your name put on something?
like a pillow, bag, ect.
I am assuming you are talking about something you order and you are opting to put a name or initials on it. This is called "Personalizing"
Tagging, is the appropriate name for placing your name on objects such as pillows or bags. It can also be referred to as labeling and when it is done by companies it is known as asset tagging or asset labeling.
Last time I had my name put on something it was called a traffic ticket.
Could be 'Branding"