You can use the Tile tool to achieve this result.
I've attached an example workflow (note I filtered out the empty rows in your sample data)
To get the output you require you need to configure the Tile Method to Unique Value and set the unique value to your name field.
Make sure to check the "Leave unsorted" box also, this will ensure the output is in the same order as the input.
This configuration will asign an ascending number to each unique value in the name column (output as Tile Number) and then an ascending count of the times that name appears (Tile Sequence Number)
The sequence number is the value you are looking for.