+1 on it being the front diff vent...

https://www.realoem.com/bmw/images/diag_18xa.png

https://www.realoem.com/bmw/enUS/sho...diagId=31_0547

added info
 

Sure looks the part, and I finally located the other end going behind the engine into the transmission tunnel (but didn't see what it connected to).

It's a vent tube alright, but how the heck did it get turned upside down and backwards? without breaking! and where is the "bracket" it attaches to? There seems to be a choice of two, according to Realoem.
Attachment 81872

I'll find something to attach it to, after I make the opening face down. I'm sure to find more misbegotten malassembled items in my search for the vacuum leak (this isn't it).

And, regarding how to get behind the engine to look for vacuum pieces:
I think I've found it; there are two bolts holding an undertray below the microfilter area. There are angled guides to help slide it back into place (I think). Added 3/25 here's a couple of videos showing how to do it: https://youtu.be/Qxo5tg1qVRc or https://youtu.be/Eq00ZAOwc88 works on both 3.0i and 4.4 engines. I'll try it Monday, after the wife gets home (I limit my undercar activity, and also disabling vehicles, to times when I can get an unwilling assistant, to lend me a hand).

One end of that tube goes into the top of the front differential.

The other end attaches (around those two radial rib things a few inches before the u-turn) to a bracket near the top of the engine oil dipstick tube. I expect if you forget about the vent tube for a moment and look at your dipstick tube you will see some clues that make it clear.

I have the same car as you - 2001 3.0i, same color too, and that's how mine is.

attached tube
 

I gave up trying to squeeze in under the X5, sitting precariously on one jackstand, resting on a broken board on sand/gravel. I dug out my "truck" ramps that had been under my trailer in the garage, and drove the X5 up onto them. Much better, feel safer underneath.

While under there, I looked around at all the "new to me parts", and was pleased to see no sign of leaks (some seepage, though, on the oil pan), no loose or missing parts, and driveshafts, guibos, and rubber boots were good. My friend that was fixing it up for me over the previous 3.5 years before we swapped cars, did a good job. I haven't found the source of my codes, but they started months after I got the X5 from him.

I did re-route the mystery tube to near the bracket that had an empty clip; that must be where it had been, so I just zip-tied it to the dipstick, instead, because my fingers weren't nimble enough to use that clip. Fair enough.
Attachment 81874 here's the empty tube clip

I put the truck back on the ground, and tomorrow I'll start afresh, looking for the source of the vacuum leak. I found out how to remove the micro filter housing, etc. from over the engine (I added an addendum to my previous post about two videos that clued me in). Smoke tester first, carb cleaner second (while the engine is idling, while monitoring the Torque Pro app on my phone), and hope for the best.
Added 3/26: homemade smoke tester failed after warm-up, the $7 soldering iron burnt-out after only 30 minutes of total usage...my advice, don't build one like mine!

a question for the Mastertech!
 

I've been delayed in working on the X5' vacuum leak mystery, by wind, weather, Dr's appointments, testing equipment failure, and various innopportune events (even my personal cell phone gets a BSOD whenever I need to use it, then works perfectly when examined by a AT&T rep, or an IT technician). And so it goes.

I was looking at another thread https://xoutpost.com/bmw-sav-forums/...-up-p2272.html, and saw this statement by a trusted source on this forum:Since I have seven more months to find/solve my vacuum leak?/SES problems, I may not need an emergency "get out of jail, free" card, but that quote interests me. What adaptations need to be reset, and is that a "last-ditch escape clause" in case I never find what is causing my error codes? Andrew Wynn, please tell me! PM me, if you want!