You would see the same thing on a treadmill holding onto rail or pushing baby carriage. You've got the only solution already, putting in pocket or pinning to clothing. The down side to that is you don't get any heart rate data for that time.
Fitbit, and all wrist-worn trackers, detect steps from the normal swinging of the arms while walking. When the wrist is held steady as supported by a shopping cart, it doesn't see any motion, so has no way of knowing what your hips and legs are doing.
Sometimes if I happen upon a cart that runs smoothly, I push it with one hand for a while.
Before posting, re-read to see if it would make sense to someone else not looking at your Fitbit or phone.
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