New version of OS X called Mavericks gives new function called Push Notifications. It's great feature, because it allow to receive notifications even if my macbook lid is closed. But for some time I started using Chrome instead of Safari, but when I click on Notifications Safari is automatically open. Is it possible to open those notifications using Chrome? Chrome is setted as default browser.
No, as it's specifically Safari Push Notifications using Apple Push Notification service (APNs).
In theory, Google could release a version of Chrome via the OS X App Store, and as a registered developer, leverage APNs, essentially creating Chrome Push Notifications (CPN). Similar to Apple, Google would have to provide their own developer console for 3rd parties to register their web site / application for sending CPNs. Doing this would allow Google to revoke anyone abusing the service. Google would host servers to receive CPNs, which in turn would forward the messages to Apple's APNs and off to Chrome running on the OS X. I don't expect we'll see Google supporting this anytime soon, as it's exclusively an OS X technology. An alternative is Google could run their own daemon on OS X (and other platforms) to receive push notifications and wake up Chrome.