Is there an Android equivalent to iOS's "CallKit"? This actually seems like a very Android-like feature. When I heard about it, I thought there might be an Android equivalent.

It lets VoIP apps use the standard phone app to call/receive calls, integrating with the phone's contacts and recent calls list. Basically, it turns VoIP calls into regular phone calls (at least for apps that support it).

If there is nothing like this on Android, is there any indication that Google will add something like this to a future version?
Android does, but it's not very well documented. The ConnectionService class was introduced in Marshmallow and allows an app to register itself as a voice service, the user needs to manually enable the voice service through the phone settings (same place you would enable/disable SIP accounts).

Once this is done you can choose between registered voice services when placing a call using the native dialer, or set one as the default. Incoming calls to your voice service should also trigger the native dialer.

Side note, but also interesting, Marshmallow also introduced the ability to register your app as the default dialer too.