Are your finger muscles and tendons interconnected?
That's a really interesting question!  What might help you the most is reviewing hand and forearm muscle anatomy.  I have some examples that could help.  The flexor digitorum profundus and flexor digitorum superficialis are hand muscles that originate in the forearm region.  At the wrist they split into four tendons (so FDP has 4, FDP has 4) one for each non-thumb finger.  They help make a fist.  Then there are the intrinsic hand muscles.  One interesting example is the lumbricals, which originate at the FDP tendon and have a very specific action. 
To answer your question, I don't see any reason why exercising the four non-thumb fingers is recommended against, other than the possibility that you could be inadvertently moving/stressing the thumb.  I don't believe there is any connection between muscles of the thumb and non-thumb fingers, other than proximity of muscle/tendons.