In theory once we add a tap for multiple outlets the cord needs to be a 12 or have ocpd like a power strip has. The 15A end will feed thru 20A on a 20A circuit but there is the chance of overloading the additional cord.
There is some safety for this by a couple methods but mostly to design and you need to bear with me on it as I am not an expert and all this is some generalization and there are or may be specific exemptions but,,,,,,,,,,,,,, we can assume for the most part we are talking about UL listed with factory ends.
You were not the genius that picked your comp, someone did this for you in some respect. The engineer loaded all the pieces to this system to their legal design limit, figuring motor factors near 13A on a 14 wire, others load it to 80%, some can be 100 for non continuous and some are calculated like lamp cord, there is some headroom for error but any one device cannot overload either a 14 or a 12 wire. You can put 2 devices of 15A each and it will simply trip the breaker but once the operator adds a cord he is sposed to recognize the fact that it needs to be as big as the appliance needs if there is one outlet like the end of a common cord, or we use a 15 or better cord. We could in theory keep adding outlets forever if the wire is a 12.
Since the common power strip plugs to a 20 theru a 15 circuit they add a breaker just like some fan that has a duplex on it, has a breaker and 16 wire or cheap 4 way with cord. A simple hard 3 way will feed thru 20A.