Contract Assets are a unique Custom Object type that are a part of a Assignment record which is associated with a Contract record. They cannot be pre-defined and behave like Free-form Custom Objects in the Contract record.
To add an Asset to an Assignment record, the Assets Custom Object must be added to the Contract record layout.
On the Contract record, locate the Assets Custom Object and click 
to add an Asset line to the record which can be "transferred" to the Assignment record. Enter a name for the Asset and save the record.
For more information, click the link for the Assignment Objects article.