Yes, I will apply a simple coat of car wax to my clubs including shafts. DO NOT apply to the face of the club that will break a rule that either alters, changes, or enhances the performance of the club.

Probably not on the face

I use regular car polish, as well. Keeps the clubs cleaner between rounds, and helps protect against moisture damage.

The setback of applying polish is that the clubhead will feel somewhat greasy. And if we touch the clubhead during play, I amfraid the hands will feel greasy too - especially when gripping.

I use Turtle Wax polishing compound and it doesn't leave a greasy feel. Being a rubbing compound, will also take care of small scratches and those dreaded sky marks on the crown of woods if happens.

I use a metal polish called "Flitz"which is non-abrasive. Careful with some of the other chrome polishes as they can be highly abrasive. Oh, and never use steel wool on chrome.
For a quick cleaning try Windex - surprising how well it works!

specifically wondering if the abrasiveness of polish will damage clubs

I just used a mild soap and water solution and an bath towel and wipe them down. The towel will make mine shine. I have my solution in a Windex bottle and spray it on the clubs and it does the job. Most waxes have some abrasive materials in them. Bet if you was yours and I wipe mine, you will not be able to tell the difference. Keeping them cleaned is the objective.

I just use warm water and groove brush and microfibre cloth to dry and buff.