Does the pool miners in mining pool receive the same block in case of a fork? For example there are 2 mining pools A and B and they mined a block in same time.

Then, A mining pool got a block that A mining pool mined?

Or nodes in mining pool receive independently so they select block regardless of mining pool?

I have known nodes choose a block which are propagated early to them and block is propagated not geographically close node but randomly (I mean block is propagated close nodes but this ‘close’ do not mean geographically close but close in network).

When miner is mining or propagating, miner can recognize which node is mining in same mining pool?
Most of the mining pools that are run are managed pools, which means it is operated by an individual or a company. The pool server runs specialized software and a pool-mining protocol that coordinate the activities of the pool miners. The pool miners need not run a full Bitcoin node as the pool server runs one and validates blocks and transactions on behalf of the pool miners. 

The pool server constructs a candidate block by aggregating transactions, adding a coinbase transaction (with extra nonce space), calculating the merkle root, and linking to the previous block hash. The header of the candidate block is then sent to each of the pool miners as a template. Each pool miner then mines using the block template, at an easier difficulty than the bitcoin network target, and sends any successful results back to the pool server to earn shares.

Since, the pool miners receives the block template directly from the pool operator, all the pool miners are building the candidate block on the same version of the blockchain.