name this.... blue thingy

That connector is for the "safety switch" (item 132 on your 400i schematic "air flow sensor switch") built inside the airflow meter (so I wouldn't remove it). The way it is supposed to work is:

1. if the airflow plate is not deflected (i.e., the safety switch is electrically closed), the fuel pump will not run if the key is "on", or

2. if the airflow plate is deflected (i.e., the safety switch is electrically open), the fuel pump will run when the key is "on".

If you just unplug/remove that safety switch, the fuel pump will always run whenever the key is "on" so for normal operation you wouldn't notice any problem, but it's there in case you have an accident -- you don't want to be unconscious inside of a mangled car with the engine stopped but the key still "on" and fuel spurting out everywhere under high pressure.

Obviously, someone has made some fairly significant wiring mods on your 400i, but if everything was working as it was designed:

engine "off" + key "on" + airflow plate in rest position = fuel pump does not run;

engine "off" + key "on" + you manually deflect the airflow plate some with a finger = fuel pump runs;

and whenever the starter motor is engaged (i.e., key in "start" position) = fuel pump runs (regardless of the airflow plate's position)

 

Don't have a clue but on mine the wire runs across the top of the engine over the cam chain cover and back along the side of the exhaust cam cover and enters a wire loom between the left side FI unit and the cam cover. From there it drops behind the engine and apparently into the passenger compartment (I think, it's dark back there!). I also have one on the left FI unit on it's left side that does not have a plug in it. I assume these are something added to be EPA compliant. They do not appear anywhere in the factory Injection manual.

 

Yes, if your fuel pumps run when the engine is "off" and just the key is "on", this is a risk -- is that the situation you have now?