Before we try to decipher what is wrong here...

How are you connected from the MOX to the MONTAGE.
Standard 5-pin MIDI cable?

Let us know... it makes a difference.

Am I missing some option in the settings or does KBD CTRL somehow unify all selected parts into 1 channel? I thought Montage controls itself using first 8 channels in MULTI mode, but it looks like there's no difference between MULTI and HYBRID.

”MIDI I/O Mode” = Multi — each KBD CTRL will receive MIDI In on the Channel corresponding to its Part Number. You transmit IN on Channel 1, only Part 1 will sound. Even if you select a Multi Part Performance your external device will play only one Part at a time (assuming you are transmitting on just one channel).
“MIDI I/O Mode” = Hybrid — all KBD CTRL Parts will play together on the I/O channel you select. If you select a 5-Part MONTAGE KBD CTRL Performance, and “Hybrid, Ch 1” is the setting, all five Parts will respond exactly as if you were playing the MONTAGE Keys. Each Part 6 thru 16 will sound individually when you transmit IN on the Part’s correspondingly number MIDI Channel.

I am using USB.

What's wrong is that master keyboard (Yamaha MOX) is transmitting notes on Channel 3, but I hear sounds from Part 3, Part 5 and Part 7 coming out of MONTAGE.

If I turn off KBD CTRL for Part 3, then MOX controls only Part 3, but if I have KBD CTRL on Part 3, Part 5 and Part 7, all 3 parts are being controlled by MOX.

Expected result:
Only Part 3 should be producing sound when I press note on MOX.

Use a Standard 5-pin MIDI cable. (You didn’t get the hint... it makes a difference).

If you connect via USB you are only replacing the MONTAGE keys with your external device. That is not a MIDI connection technically speaking... connecting directly to the USB Port is a way in which you can use an alternate controller with limited options (like a Roli) in place of the MONTAGE keys... for what you want to accomplish — Use a Standard 5-pin MIDI cable.
Set [UTILITY] > “Settings” > “MIDI I/O” > Set “MIDI I/O” = MIDI (green).

If I turn off KBD CTRL for Part 3, then MOX controls only Part 3, but if I have KBD CTRL on Part 3, Part 5 and Part 7, all 3 parts are being controlled by MOX.

Yep, that’s correct. It behaves exactly as if you were playing the MONTAGE keys. Because you are substituting the external USB device’s keys for the MONTAGE keys. The “MIDI I/O Mode” settings do not apply in a situation where you are using the USB “To Device” port.

Use a Standard 5-pin MIDI cable. It makes a difference.
If you wish to use the Advanced “MIDI I/O Mode” settings you must not connect to the “To Device” Port.

Thank you very much!

Issue is solved by using 5-pin midi cable instead. I finally have my idea of perfect 2-keyboards setup working. 😉

To be more specific, it matters which USB port is used.

The USB TO DEVICE port can accept USB-MIDI as of a recent firmware update for both MODX and Montage. Using the USB TO DEVICE port has different rules than other connections. USB TO DEVICE is like extending your local MODX piano keys to an external device using a USB cable. And you know that the local MODX piano keys work the way you describe - all Parts with Keyboard Control turned ON will sound together when you strike a local piano key. USB TO DEVICE connected keyboards will act the same way.

USB TO HOST connected devices (going through a computer, tablet, or other intermediate device) will follow the same rules as the 5-pin DIN connector (in MIDI mode, not USB mode) with respect to MIDI channels and how they interact with Keyboard Control settings. USB Port-1, in USB mode, is what targets the tone generator and USB Port 3 targets the 5-pin DIN connector. In USB mode the 5-pin DIN ports no longer are connected to the internal tone generator directly as the 5-pin DIN port is directly connected in "MIDI" (not USB) mode. Even in USB mode, the 5-pin DIN can be routed back to the tone generator by using MIDI routing rules on your intermediate tablet/computer device through its software configuration.