From the ‘Share & Assign’, you can Assign Owner for the selected record.
You can assign any record to specific user or team by clicking on “Assign Owner” option from specific user’s details.
On clicking, you will have the option to add record either in existing marketing list or create the new one. 
After selecting the marketing list, click on the Add button to complete the addition of records to the marketing list.
Note: Assign to marketing list icon under map section, will assign all records of particular entity to the list. Also, it will get assigned only when records of single entity are plotted on map.