Are bitcoins (or fractions of bitcoins) labeled with IDs?
So are bitcoins, or the smallest allowable bitcoin fractions, distinctly labeled entities?


They are unspent amounts (outputs of prior transactions) associated with addresses. Unspent transaction outputs are abbreviated UTXO.

Bitcoins or fractions (millibits, Satoshis etc) don't exist as entities in computers running Bitcoin software. The computers just keep track of the movement of amounts, they keep track of UTXOs, transactions, addresses and so on. Those are the identified entities.


  what would double spending actually look like in the bitcoin system?


Two transactions with the same input UTXOs.

The first one to be mined and "confirmed" will be valid, the second one will be regarded as invalid. There can be a period in which the outcome is uncertain until a consensus emerges.

See also Replace By Fee (RBF)