Not gonna lie here, that looks way more like an disadvantage than an advantage.
The main problem here is that this seems like an unnecessary redundancy. Yes, morays have 2 sets of Jaws, but the secondary set is inside the mouth and helps pull prey inside. In this case, rather than help, the second jaw gets in the way.
Problem 1: To open the lower set, you only need to open the second mandible, but to open the first set, you have to move the entire second mouth arrangement, thus giving it a harder time for closing, which will require more muscle. But looking at it, the upper jaw space for muscle attachment is divided between it's own muscles and the muscles that support the jaw below it, meaning it will either be weaker or will be bulkier to allow it to work somehow. In addition, if only the lower jaw it's broken, then it might be an advantage as it can still eat in a way,but if only the first one is broken, you'll likely have the entire system compromised. It might be able or eat maybe, but don't trust that "maybe" too much.
Problem 2: weight. We're talking about having such an arrangement, we're talking a fair amount of extra bone, muscle and weight to the skull. If this thing supports its head like dinosaurs and most quadrupedal animals, it'll need to make the skull lighter through whatever means it can find, likely meaning a small brain.
Problem 3: this design doesn't seem to give you the advantages you said. I've already talked about how one jaw will always get in the other's way, so unless you have a very specific scenario, I doubt it'll be able to land a double bite. Moreover, the fact both have similar dentition seems to indicate both are used for eating. In that scenario, if you try to talk with a mouth while eating with the other (which will already be hard in itself), you'll either choke, cause your larynx will be open for the incoming food, or you'll stop talking for a bit until the food goes down and the larynx opens again.
Now for your suggestion: separated them altogether, with one being used specifically for talking and never for biting. That actually sounds like something that could maybe work out, considering it never has food or anything else shoved inside this breathing mouth, or else it'll choke to death as well (meaning even this scenario has its disadvantages). Additionally this mouth might not produce normal saliva, as there's no need for it. But still I highly doubt such a small benefit would be selected with so many other problems.
So summing up: could this happen? Maybe. Are there problems with such an anatomical choice? Yes, a lot of them. The skull overall will be heavier, both mouths compete with one another with the upper one seeming to be at a disadvantage (both for eating and talking scenario), there will be problems with muscle that'll likely result either in 2 even heavier but moderately strong Jaws and 2 weaker Jaws than if there was only one, there might be a higher risk of choking due to a larger opening in case a mistake is made (one for eating and the other for breathing scenario) and possibly others I couldn't see. So overall this design doesn't sound advantageous to have. It sounds better to stick with either just one jaw or 2 Jaws independent from one another.