With all of the recent updates to LR Mobile and the CC, does LR now have the ability to share a catalog between two computers?
Sharing a catalog between two computers is not a LR Mobile or LR CC issue.
The catalog can be placed on an external drive and moved between computers as needed. That's pretty much the only way to do it.
Some people have tried to put the catalog on a cloud sharing service like DropBox, but this is fraught with danger, if you open the catalog file before it is completely updated by DropBox, you will wind up corrupting the catalog file on both computers.
I also know that I can drag images with me on an external drive in a catalog, and then update my desktop when I get back home ... but that's work.
You could create smart previews and then you wouldn't have to "drag" images with you. But there's no way around doing actual "work" to make this happen.
If you put your catalog on an external hard drive you can move it back and forth between computers. BUT that also requires all the images to be on the external drive. Another option is to put the catalog, and/or images in a dropbox folder that is synced to both computers. Or, you can create smart previews of the images you want to work on with your laptop. Put the catalog in dropbox or an external drive, and leave the full res images on your desktop.
Yep, I know that I can drag the catalog to an external device and bring that with me. I've been doing that for years.
I'm wanting to do it via the Cloud, like I can on my iPhone or iPad, without having the headache of copying data.