No, .bit domains are apart of Namecoin and the resulting domain names resolve like any ordinary domain name. You need a Namecoin client to use them. There was a bit of publicity for it when it first came out but Namecoin isn’t used very much nowadays. Additionally, Bitcoin addresses are intended to be single-use, and the developers don’t want to implement features that would encourage users to make the mistake of reusing addresses.
The Ethereum addresses by comparison are something that (from my understanding) developed alongside the main Ethereum developers and included as a default feature in Ethereum (rather than an altcoin). Additionally, Ethereum addresses are meant to be reused, making such a shortcut more appropriate.