What's this part good for on my exhaust?

Neither do I, and I did say exactly the same, in fact.
That doesn't make then necessary, anyway.
Aside from my own boats, I can think of several other builders who went the opposite route.
No right or wrong here, as long as the final result ticks all boxes.

Yup, I believe that's negligible, for two reasons.
First, even if dry exhaust pipes can indeed get much hotter than the engines, in terms of volume and mass they are just a fraction of them.
Secondly, while diesel engines are running, the amount of air exchanged inside the ER is massive.
Just take the OP engines, with their 24+ liters displacement each: when they are both running at 1800 rpm, they suck (and spit through the exhaust) well above 60 cubic meters of air per minute!
That's a HUGE amount of fresh air (well, as fresh as it is outside, of course!) constantly flowing through the ER, and is also the reason why normally the problem of cooling the ER is only relevant after the engines are turned off, not the other way round.
And obviously, with the engines off, the exhausts cool down MUCH faster than the engine blocks, due to their sheer mass of iron.

Nope. A bit counterintuitive maybe, but it's actually the other way round.
Raw water is re-injected only when the engine is running, and while its quantity increases with rpm, volume and speed of exhaust gas also increases, and more than proportionally compared to the water flow.
As a consequence, water never stands a chance to travel backwards at high rpm, regardless of where and in which way it's re'injected into the exhaust.
So, it's only when idling (or at very low rpm, like upon maneuvering or at low displacement speed) that water reversion can realistically be a danger.
And by dumping the water on the lateral or upper side of the exhaust pipe, the relatively low water flow is better mixed/nebulized with the exhaust gas, which in turn pushes it out more easily.
By re-injecting water from underneath, part of it can progressively "stagnate" along the pipe bottom, particularly when not slanted downward, as in this case. And eventually, maybe with the help of some boat movement, this water can indeed slosh backward.

Coming to think of it, I understand the doubts on this matter, because marine exhausts are tricky, and rarely well understood also among professionals.
So, having time to kill (courtesy of the lockdown ), I'll try to better explain what I meant, when I said in my post #7 that there's more than one way to skin this cat.
Not one of the most exciting boating topics arguably, but I'd be interested in additional and/or contrary views from anyone who will bother reading the following.

Now, leaving aside the old school dustbin silencers with transom (non u/w) wet exhausts, to my knowledge there are essentially two alternatives for water injection in u/w exhausts.
Which nowadays are by far the most popular solution with planing boats, because on top of working well for silencing, they also reduce backpressure through scavenging effect, above a certain speed.

The first is by placing the re-injection downstream of a relatively long dry section, and just before any rubber or GRP bit which could not withstand the temperature of dry exhaust gas.
The exhaust system of which the OP posted a pic in post #11 seems built in this way, assuming that the mixer point is the one that we guessed to be.
But normally, exhaust systems built in this way have a first section upward slanted, going as high as allowed by the ER headroom, and the re-injection point is placed along a downward slanted section that follows the first, before the hull connection.
The two pics below are examples of this setup.
In the second pic, the steel elbow connecting the turbo to the vertical pipe is missing, but forget that.
I'm using it because it shows the re-injection point along the downward slanted pipe, just before the rubber section that connects the pipe to the hull GRP box.
It's exactly in the same position also in the the boat of the first pic, but not visible from that perspective.
I think it's easy to see why with this arrangement makes water reversion much more unlikely, compared to an arrangement like the one in post #11, where the water doesn't need to "climb" uphill to go backwards.
And as far as can be told from that pic, there was enough ER headroom to raise the two first vertical sections further, and then slant them downward towards the injection point, but now I'm just speculating, of course.





The second alternative is by placing the re-injection point as close as possible after the turbo, almost eliminating the dry section of the exhaust.
I am attaching below one example of this solution, which several yards began to use upon the adoption of V-drives, for position/geometry reasons.
In this case, when applied to V engines, it's necessary to split the water re-injection into each bank.
Besides, again in order to prevent water reversion, these re-injection pipes must go a bit higher than the upper part of the exhaust pipe, before connecting to it. This way, when the engine is turned off, the water still present in the pipe doesn't drain into the exhaust.
Besides, the very first exhaust section attached to the turbo flange is actually jacketed inside, in order to cool down the pipe as early as possible, but forcing the water to mix with the exhaust gas further downstream from the turbo.
This is imho the most elegant solution, that btw doesn't require insulation anywhere and leaves the whole exhaust visually accessible to immediately spot any leak or corrosion.
But it's practically impossible to retrofit it in a boat where the hull connection is not suitably aligned.
So, that was not feasible in the OP boat, for instance.

Yup, all agreed. I didn't see your post 'cause I was writing mine.

I was also wondering if that boat is on V-drives, because she does appear to have a V-drive gearbox, but the position of the exhaust along the hull would be more astern than usual.
I'd be curious to hear what's the boat which we are debating...

Absolutely. Expensive they are in absolute terms of course, but if badly designed, their overall cost can be a multiple, potentially stretching all the way up to a complete engine rebuild.

Which I'm wondering if by chance is also the reason why the OP had to rebuild a couple of engines which MAN introduced only 10 years ago, so they couldn't be older than that. Also because he specified that the request to rebuild the exhaust came from the MAN dealer...

Oscarvan, I'm not sure to understand your point.
I mean, I'm also aware of engines killed upon startup, after sea water filled some cylinders while they were not running, but for reasons that had nothing to see with the mixer and its placement.

For instance, some early Ferretti 175 suffered sea water ingestion while rolling at anchor with the engines turned off, because the original exhaust system had not enough drop from its highest point towards the u/w outlet, and sloshing sea water could find its way back inside the engines.
Particularly into the external cylinder bank as I recall, but that's irrelevant in this context.
The main problem was the very low headroom inside the e/r - a constraint they had to live with, once the boat was in production.
But eventually, after replacing under warranty two pairs of MTU V12 183 (whose cost wasn't trivial!), they found a way to modify the exhaust and prevent this risk. In fact, one of their dealers told me that they also made a recall of all the already built boats, to fit those modifications.
This had nothing to see with the internal raw water circuit anyhow, let alone the mixer placement.

In which way were the engines that you have in mind flooded by the internal raw water circuit while not running?
And what had the mixer position to see with that?
I'm not saying it's impossible, mind. If you say that it happened, I take your word for it.
But I'm genuinely curious to understand how.