What distinguishes a miner from another? So I've been recently studying about Bitcoin and had few questions:

1) What distinguishes a miner from another miner? Is there any proof that two miners are not the same person?

2) What can we say are the transactions accepted by a miner? Is it all transactions in the blocks after itself, that have been added to the miner's blockchain branch?

3) In any case, what's the proof that a transaction has been accepted by any miner?

Thanks!
You can't find out whether 2 blocks have been mined by the same entity (e.g. person) for sure. Guessing is possible. If you go to http://blockchain.info, you'll see a "Relayed By" column in the table right on the start page. You can be reasonably sure that that mining pool mined the block. Some mining pools always send their money to the same address. In that case, you can be even more sure that all the blocks whose coinbase transaction went to that address have been mined by the same entity because else they'd be giving a lot of money to a different entity.

The proof that a transaction has been incorporated into the blockchain is that there is a valid blockchain which contains a block which contains that transaction. You can't simply take a blockchain and swap a transaction in it out for a different one because that'd make the blockchain invalid as the hashes wouldn't check out.

Furthermore, if you want to be entirely certain that the transaction isn't just in any blockchain but also in the longest one, you need to be connected to the Bitcoin network so you know the longest blockchain. The longest one is the one which counts.