more number of coats means longer protection?

It can also make it worse as the thick base cracks and peels. I would thin the first coat, sand, then apply two more, sanding between.

I agree, three coats. The advice I've seen is that three is better than two, but nothing that more beyond that helps, though it might, but I'd bet it's diminishing returns even if it does.

But I have heard that the shining, glass-like coating on piano consists of ten or more coats. True?

I was going to mention violins and other musical instruments. The hardness and the smoothness of the coating affects the sound.

Very thin coats rubbed down with increasingly finer grit from 400 to rotten stone. The hills and valleys of each coat get smoothed out.

That is sprayed instrument lacquer and hand rubbed between. The thickness of each coat is much less than any poly.

There are a number of coats but probably not as many as ten; the Steinway finishing tours on utube and the like don't specify how many.

That may be. I've seen custom cars with 10 or 15 coats, but that is not the same a brushing on coats of polyurethane. They use very thin coats of thinned lacquer, sprayed, not brushed.

True, but irrelevent, as pianos are very much not exposed to the elements, and the finish is a nitrocellulose based lacquer, not polyurethane.

I was thinking similar. The OP said outdoor stuff, which is a different environment. More coats may help, but I'd bet there is a significant diminishing return effect after three. Also, is the goal to protect the wood and what it looks like doesn't matter? Or must it look good too? With ten coats you might find that it protects the wood longer but starts to look bad about the same time as 3 coats.

I think one would find that due to the seasonal expansion and contraction of outdoor wood, regular polyurethane (regardless of the number of coats) will not survive more than a couple of seasons. UV exposure is another hazard for regular polyurethane.

Spar varnish, on the other hand is designed for exposure, so it is softer and tends to stretch rather than break as the underlying moisture content of the wood changes seasonally. Still, one would need to recoat periodically.

Personally, weathered gray wood is good enough for me :-)