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You need to contact whoever installed that and tell them to come undo the code violations and do it right or you will call the plumbing inspector to look at it and you will ask him why the tile man is doing plumbing work without a license.

The solution is simple but it depends upon which code your under because the IRC is different from the UPC.

Would this be the proper way of doing it?

If the horizontal center to center distance between the two drains is 30" or less you can hook up up that way. In fact you can get a kit for a kitchen double sink and use that.

48" is too small for a double vanity. The smallest recommended size for a double vanity is 36" in between the center point of the two sinks, which translates to a 60" space if you want any room between the edge of those sinks and the wall. Two people cannot use the sinks at the same time in 48" so you have not really increased the utility of the bathroom at all. In fact, you've probably just created a red flag situation if you ever go to sell, even if you bring the plumbing up to code.