Is it possible in future bitcoin splits in multiple groups of nodes/blockchains and every group claims as the original one like we see in religions and their sects.
Yes, a split is possible, and is called a hardfork. This has happened before, but AFAIK there's never been different groups claiming both branches should continue: rather, a code change in the core client causes a fork, and there is
consensus on which chain should continue.
Sustaining a split would probably mean that Bitcoin Core is forked, so we have more than one maintained reference implementation: each chain might still like to call themselves "Bitcoin" and claim themselves as the "true" child of the original vision of Bitcoin, but by necessity one or both will have some identifier added to the name it's popularly known by.
Note that transaction outputs from before the fork would be spendable in both chains, while those after the fork would be spendable in only one chain. The effects of this would be interesting...I think it'd be highly destabilizing, discouraging the sustained use of the smaller of the two forks.