Press sauté. Add beef. Brown beef until no longer pink. While beef is cooking, grate zucchini or shred into zoodles.
Press cancel. Add beef to zoodles and wipe out instant pot. Mix browned beef, zoodles, eggs, parmesan, 1 cup mozzarella, salt, pepper, and marinara in a bowl. Add 1 cup water to instant pot. Add trivet.
Divide beef mixture in two portions. Line each of two stackable pots with parchment paper. Cut parchment into strips and line sides of each pot. Add half of beef mixture to each pot. Spread half of sour cream, half of chives, and half of remaining mozzarella to each pot. Stack pots and seal lid. Lock lid of instant pot. Set valve to sealing. Program for manual-high and 12 minutes. Utilize natural pressure release for 10 minutes. Carefully move valve to venting. Remove pots and serve.