Is 200x magnification realistic for a kids' microscope?

That is the claimed maximum. Toy scopes at this level usually give usable detail at lower zoom and get blurry near the top, so expect clear views of insects and leaves rather than cells.

Does it work with any phone?

It has a holder that clips a phone over the lens. Fit depends on phone size, and the phone camera does the viewing, so no app is needed.

What batteries does it need?

Batteries are not included, so check the battery type on the box before first use and keep spares on hand.

Is it good enough for a school science project?

For simple observation of everyday objects, yes. For prepared slides or fine detail, a proper kids' microscope with a stage and focus knobs is the better tool.

How does it compare with a standard kids' microscope?

It is smaller, cheaper in build and phone-friendly, but it lacks a stage, slides and precise focusing. Choose it for portability, not for lab-style work.