Pocket door frames are usually secured on the bottom through steel straps connecting the bottom rails. We can't really see them in your photos, but I suspect they're there.
If they're not, here are a few ideas. It doesn't need to be complex.
Run long screws down through the bottom rail. You might need to pull the drywall to do this accurately. One on each side penetrating the subfloor completely would be enough.
Run toenail screws through the bottom rails. These could go right through the drywall as long as you stay below the top of the base trim you intend to use. I'd use two on each side.
Use L-brackets. One technique would be to mark the frame location on the floor, then spread the sides and install one leg at the frame location (ultimately under the frame), and finally fasten the other leg to the front edge of the frame. You could also just install them inside the door slot. Keep them back a few inches so they don't show.
If you're asking whether it's normal for such a frame to not seem very robust, they're all like that. There's virtually no load applied to the side rails, and once your drywall is installed it'll be fine.
Word to the wise: Use no longer than 1-1/4" drywall screws, and remember to only use short nails when installing your base trim. I've punched nails into several expensive oak doors in my day. It's not a fun discovery.
