Wagner are essentially a parallel army that's outside the normal command structure - the Kadyrovites are the same. Putin is still at the top, but the rest of the hierarchy is absent (which in theory makes them more flexible and makes it easier to learn what is really happening).

Back in World War II this was sort of done with the NKVD though their units were put in the Red Army when it was necessary (most famously at Stalingrad). Germany's SS was likewise an army outside the usual Wehrmacht command structure - gaining more power throughout the war as Hitler distrusted his generals.

The reason they are in fierce competition is because they're trying to win the same resources from the centre. The army has struggled to get these resources because its leaders aren't part of the ruling clique (who are instead drawn from the "siloviki" - high ranking FSB and other security services).