I want to add up the square footage for each building floor and enter the total into another field within the same feature class. Each floor has a different size and is a separate polygon with a unique ID. The feature class stores multiple buildings.
You can compute the area using a right-click on a field > calculate geometry.
Then you summarize your attribute table using the building name as a case field : you can then ask for the sum of the area.
Finally, use a table join (join by attribute) the have the total area of each building, which you can copy if needed in a new field.
If you don't have an attribute with an identifier of your building, you can try with dissolve, but it will not work if some buildings are adjacents (they will be considered as one single building).