You mean private key is not part of block chain blocks. Private key is a part of wallet which are using to store bitcoin by us. Then how we can validate particular transaction comes form which wallet.
Anything signed with a private key can be verified with the corresponding public key.
I suggest you read
this article about asymmetric cryptography.
This should let you understand the whole private-/public key infrastructure.
Is it possible to transfer one wallet btc to different wallet btc like we can say coinbase to blockchain wallet?
You can transfer BTC's from any address to any other address. This does require an on-chain transaction, of course.
Either between online wallets (what your question is) or to any other wallet (e.g. desktop-, mobile-, hardware- or paper wallet).