The bike shown in the first image is Specialized Sirrus X. You can make out from the image of the manual that it says "spacer/axle/bolt specifications". Searching for this yields the following Specialized bike manual which is similar:
https://www.manualslib.com/manual/1974494/Specialized-Epic-E... -- there are some notable differences, but the Specialized Sirrus X manuals that are online aren't in the same style.
The prior page (8) shows "SEAT COLLAR 4mm HEX" and, based on looking up seat collar in an image search, the part in question matches.
In terms of the toolbox, note that it only identified the location of the Allen wrench set. The advice was just "Within that set, find the 4 mm Allen (Hex) key". Had they replied with "I don't see any sizes in mm", the conversation could've continued with "Your Allen keys might be using SAE sizing. A compatible size will be 5/32, do you see that in your set?"