[edit: sorry, my mistake: the market cap reported is not actually outstanding shares times price, pointed out below]
You are right that there are two public tickers for Alphabet, GOOG and GOOGL, however the market cap you see when you search for their ticker is for the whole company, not the class alone.
Also, Alphabet has 3 classes of shares. A (GOOGL), B (privately held), and C (GOOG)
I assure you, the market cap is roughly $870B as of this writing.
Actually, market cap is outstanding shares times price. It is a value per company, not per class/type of shares traded. Sort of like how the same CEO would be listed under GOOG or GOOGL. You'll see the same market cap listed under GOOG or GOOGL.