Is there a WordPress plugin that counts unique visits to a website's posts and pages?

Yes you may use SEO yoast plugin which is the best for se, the price around 65 US, but you may purchase from the following site at less than half price skilldigi.com or evato

I think, there is no such  plugin but http://statcounter.com/wordpress.com/ may help you.

The best solution I can suggest I using Google Analytics along with the plugin GA Dashboard for WP.

Pros:

• For those who like it: It's Google Analytics
• Reduces load on your server, common stats read/write are database heavy
• You don't loose you old stats if one day you decide to change something or the plugin you are using is no longer maintained

Cons

• For those who hate it: It's Google Analytics

https://wordpress.org/plugins/page-views-count/

https://wordpress.org/plugins/powr-hit-counter/

yes for visitor counter  Mechanic Visitor Counter Plugin is use mostly