Shore Power

If you mean the two Hager components in between the two breakers, yes I do. They are called "contactors", and they work as follows:
When the so-called "control contacts" (A1 and A2, with the smaller wires in the center) are powered, the coil electromagnet inside closes all pairs of upper and lower contacts (1 to 2, 3 to 4, 5 to 6, and - you guessed by now - 7 to 8).
Both these contactors are of NO (normally open) type, which means that when A1+A2 are NOT powered, all contact pairs are disconnected.
And both are rated for up to 63A on the main contacts (1...8).

Now, it's impossible to tell for sure from your pic alone, but I suspect that they are wired to handle some kind of automatic connection/disconnection between EITHER shore OR genset, avoiding the risk of clash between the two power sources.

There is one very weird thing though, which I didn't notice before:
These components are often used to switch AC power by mean of a DC signal, so I've seen them used with 24V A1+A2 control contacts.
But yours are actually the version designed to work with 230V!
This is not so relevant for the main contacts 1...8, but they are EU components anyhow, NOT designed for 120V.
Now, don't ask me what happens if you feed 120V instead of 230V into A1 and A2. After all, they just control a simple electromagnet inside, which MIGHT be tolerant enough to work also with 120V.
But it's an improper usage anyway, that's for sure.

I don't think it's that simple Pascal, because 3 of the 4 main contact pairs appear to be used (large-ish black+red+white power cables), and pretty sure this ain't a 3-phase boat.
That's one of the reasons why I guessed that the two contactors are interconnected, in order to allow either shore or genset to be active, but never both.

BTW, what's the colour coding on your side of the Pond? Black/white/red make no sense in EU, for AC wires.

Aha, I see.
So, considering that the control contacts A1 and A2 appear to be fed by black and red wires, they might be meant to close the main contacts when both hots are available?
I must admit that I can't figure the logic anyway, also because if so they would only work with both shore power connected.
The output of single phase US gensets is on just one 120V/60Hz hot+neutral line, 'innit?

Good point, captholli.
That could also explain the higher absorption that trips off the 0.5A breaker.
Unless the coil is actually wired to work at 240V through a pair of 12oV hots - unusual as it would be, for my own experience at least.

Anyhow, we are now doing doing a lot of guesswork.
Lacking an electrical scheme, I think there's no replacement for an electrician checking what is what, methinks.
Asking him also to make a drawing for future reference, while he is at that.

I see. Every day is a school day, thanks!
Though coming to think of it, I guess it makes sense to keep full consistency with the shore connections arrangement.

I couldn't agree more ref. preferring a plain vanilla manual selection of AC source, if that's what you mean by manual transfer.