Do I need to anchor the new cultured marble vanity top I’ll be installing in my bathroom?

I would definitely secure your new vanity top to the underlying cabinet, and construction adhesive is one way to go. You could also use the same kind of caulking you’ll be using to seal the top. Caulking makes a great medium strength adhesive for jobs like yours. Just a few blobs here and there on the upper part of the bathroom cabinet before you nestle the top in place will do the trick. Although it’s true that the pipes might hold the top on their own, you really can’t tolerate any movement of the top at all. That’s because the backsplash of the vanity top and any ends that meet neighboring walls should be sealed to prevent splashing water from leaking behind them. Movement would almost certainly break this seal.