Does each bitcoin have a serial number? I was recently told that each bitcoin has a serial number and can be uniquely identified. Is this correct?
No, each bitcoin is not uniquely identifiable (there is no ‘serial number’). In fact, on a technical level, a ‘bitcoin’ does not actually exist as most people imagine it to. 

What I mean by that is that all ‘bitcoins’ actually exist as ‘unspent transaction outputs’ (UTXO). Each UTXO is a record of where the bitcoins reside: it will specify a bitcoin address, and an amount of bitcoin. 

For example, suppose you have 2 addresses, each with one bitcoin in it:

Address A: 1 bitcoin

Address B: 1 bitcoin

Each of these addresses will have a UTXO associated with it. Next, you send a transaction, that spends both of those UTXOs to a single address (Address C, we’ll ignore txs fees for simplicity). So now you have:

Address C: 2 bitcoins 

Next, send a transaction that spends the address C UTXO, with an output of 1 BTC to each address D and E. So now you have:

Address D: 1 bitcoin

Address E: 1 bitcoin

So now the question is: what can we say about the bitcoins (UTXOs) for address D and address E? 

We can trace the history through the blockchain, and know that the address D and E UTXOs have a common history with the address A and B UTXOs, but we cannot say that ”the bitcoin in address D is the one that was in address A”. This allows bitcoins to be fungible, preserving privacy and usability. 

Hopefully through this you can see how the bitcoins are not track-able digital objects being passed around, instead they are UTXOs that can be combined and split endlessly, as the user so desires.