What's this part good for on my exhaust?

Looks like a water injection elbow .
Show a pic of the part in situ ?

This, it's the mixer elbow. It mixes raw water with the exhaust to cool the exhaust so the exhaust temperatures don't melt your rubber hoses or fiberglass piping, from the risers to the transom where the exhaust exits the boat.

And it looks like a new one.

You got new exhausts with no mixer elbows? Sounds fishy. I'm assuming you didn't convert to dry exhausts. Post some pix.

If your previous exhaust had those mixer elbows, you obviously had a wet exhaust - more than likely with a main u/w outlet, plus bypass.
And since a conversion from wet (particularly with u/w outlet, which is the more common) to dry exhaust is akin to brain surgery, it's practically impossible that whoever rebuilt your exhaust did not include somewhere a raw water re-injection of some sort.

This is a cat that can be skinned in several different ways, though.
If you post some pics of your new setup as SeaLion suggested, it should be possible to understand what was done.
But you should show us the whole piping, all the way from the turbo flanges to the hull connection.

Btw, why did you rebuild the whole exhaust?
If your photo of the old mixer elbow is anything to go by, it was in excellent conditions!

So if that's your old mixer, its in excellent shape. Did they leave that on the boat for you? They may have replaced the entire system with new, and recognizing that mixer was in such good shape they left it for you?

What are the circumstances? Are you personally replacing the mixers, has it been done by someone else?
My guess is that the mixers were ordered from a vendor and they weren't completed. Certainly, if they're not identical; they weren't made from a supplied example.

The OP said in post #5 that the photo shows his "previous exhaust", and that he now has a completely new one with no mixers.
So, while I would also be curious about the circumstances, only with some photos we have a chance to understand what was done exactly.
In fact, aside from a very unlikely full conversion to dry stack, a new exhaust with no mixers anywhere would be a disaster waiting to happen PDQ!

Maybe the mixer is the SS between the black and the blue? That would make for a long hot & dry section (black) but maybe necessary for a V engine? Looking forward to the YF expertise on this. Any chance you have a "before" pictures?

Is the black, a insulated fiberglass wrap over the exhaust pipe? It should be. If so it's fine, you should have the mixer all of the way over on the far side just above the blue silicone hose where it is injecting water there into the exhaust. I can see a shiny S/S piece and might be the mixer.

Ask them where they are.

Not fan of water injection from underneath, would move it above centerline of the horizontal tube.

So they made you replace that mixer in the picture? WOW! Unless there is something wrong internally that we can't see....

I'm sorry to say that I don't like what was done in more ways than one.

1) that type of insulation on the (very long!) dry section makes visual inspections of the steel pipe impossible, and I would have asked removable wraps instead. But in fairness, this is a sort of pet peeve of mine, because that method is actually increasingly popular, also among OEM installations.
2) The "Y" connection is made in such way that the exhaust gas have a much shorter and straighter path in the external cylinders bank compared to the internal one, and this is bound to create a difference in backpressure. In this case, I very much doubt that the OEM installation was similar. But if it was, I'd be curious to hear what is the boatbuilder.
3) For the s/steel plug which might be meant for raw injection, aside from agreeing with PacBlue, it looks rather small. Besides, I see no reason for fitting it in a section of the exhaust which looks pretty much horizontal. It would have been much more logical to fit it above the last curve, and pointed downwards, to avoid any potential risk of water reversion at idle.
4) Lastly, I can't see any low rpm bypass, which engine manufacturers always require with u/water exhausts.
Actually, it's possible that the bypass is included downstream of the blue flexible connection, inside the GRP box, but that's something I would want to check.

Anyway, if the engine were rebuilt by a MAN official dealer, they should give you a warranty.
And for the validation of that warranty, they surely want to check and approve the whole exhaust system, both statically and upon seatrial.
There is a specific MAN protocol for that, that they should complete with all the relevant data and hand over to you, duly signed.

I see that you already released the payment, and I appreciate your aim of supporting the company that did the job, of course.
It would have been reasonable to wait the engine dealer approval, though.
At least in principle. On the other hand, in these lockdown days, normal principles are pretty much going out of the window....
All the best for the job completion, anyway.
If nothing else, I would expect that your supplier will appreciate your payment, and remain available for any additional support that the MAN engineer could require.

Nothing wrong in a long dry section per se, but necessary, it definitely isn't.
In my V8 MAN installation (as well as many others that I've seen) the dry section is negligible.
I mean, like 10 inches or so - if that.

All great information. That's why I was curious to see a "before" pic. I figure the longer the dry section, the more ER cooling you need, but maybe that's negligible.