So are bitcoins, or the smallest allowable bitcoin fractions,
distinctly labeled entities?
No, but almost. Bitcoins are literally just a value which is assigned to a script (see Scripts below). There are many consensus rules that determine whether that value is valid bitcoin. The most important of which is that It must be from a valid unspent transaction output. This is labeled, by a
Bitcoin uses a TXO (Transaction Ouput) model instead of a account model. This means, a transaction always has at least one input and one output (but can have multiple).
Inputs vin Outputs vout
1 BTC to Address 1 0 -> 0.999 Address 2 0
-> 0.0009 Address 1 (Change) 1
Note: The leftover
0.0001 btc can be claimed by the miner as a tx fee.
Let's say that vin 0 references a txid and vout which has already been referenced in a previous transaction. If that is the case, then that txo has been spent (the entire blockchain is checked for unspent outputs), and that transaction would be considered a double spend, and rejected by the Bitcoin network.
An amount of bitcoin is locked by a specific script. The script can represent a public key (P2PKH - Pay to Public Key Hash) or multiple (multisig P2SH - Pay to Script Hash), for example. But, the spender has to be able to satisfy the script in order to spend the bitcoin. Usually the script requires signing the transaction with a private key.