Re-plumbing kitchen sink
It could possibly be a clean out valve. These valves are often found under sinks which clog easily and/or often (like the kitchen sink), because they provide an easy way for a plumber to access the clog. It is much easier for a plumber to access a clog in a drainpipe that has a clean out valve, because the plumber can access the clog through the valve, which is usually a straighter line to the clog than having to go through the sink drain's p-trap, where drain cleaning equipment might become stuck or at least more difficult to maneuver.
I would not recommend routing a second drain line through the mysterious valve for three reasons:

You would need to ensure that the new left-hand sink drain had a proper p-trap, similar to the one currently under both sinks, as a p-trap helps prevent clogged drains and keeps sewer gasses out of your home. Additionally, many localities require p-traps under sinks or other places where sewage gasses might come up into your home.
If that mystery valve is a clean-out valve, you will be eliminating or at least restricting the point of access to it, and if your plumbing ever gets clogged behind that drain, you (or a plumber) will have a much harder time accessing and unclogging the drain.
Unless you know for sure that that mystery valve is in fact connected to your main drain line and is in good working order (no clogs, leaks, or unsound pipes), you might be allowing water to leak into your wall, which could lead to mold or other structural damage.

I would just leave your kitchen sink plumbing as it currently is, since it appears to be plumbed correctly, and bubbles coming up on the other side of your sink is a common and harmless problem. I often experience that problem myself, and the way I handle it is by simply running water down the drain in the sink where the bubbles are coming up. Usually, that washes the excess suds down the drain, and it is a simple, effective solution.