I'm trying to find out how much a single "strand" in a ribbon cable can carry. By "strand" I mean the red strand or the green strand in a multicolour ribbon cable. An example of the product: Multicolour ribbon cable
(I look at the table: AWG wire sizes used in flat ribbon cable, the 26AWG row.)
Out of no actual knowledge, but 2.2A seems high to me.
The application needs to drive 12 solenoids which consume 1.2A at 12V per solenoid. Usually only one solenoid will be driven at a time. Each will be activated for less than a second.
Current ratings are really thermal ratings. Temperature rise in the conductor can reach a point where the insulation is damaged. Temperature rise is not instantaneous of course.
Ratings are normally specified for worst case conditions.
In your case, if you have a duty cycle that is some small percentage of 100% then you probably can achieve the max rating easily. That is assuming that the ambient temp is in a typical range, and not enclosed in a cabinet which might reach 70C.
Rule of thumb solution to both temperature rise and voltage drop:
You are powering a solenoid, which is a coil made from a long thin wire. Being a coil concentrates the heat in a small space. If you use a supply wire that is much shorter and a bit thicker than the wire the coil is made from then it will be fine.
If the temperature rise in the coil is okay then the rise in the wire will be less so will be ok too.
If the supply wire is short in comparison to the length of wire in the solenoid then it will make little difference to the current supplied to the solenoid.