Are bitcoins (or fractions of bitcoins) labeled with IDs? Many sources describe the double spending problem by saying something like "people cannot be able to spend the same bitcoin twice." [1][2][3] So are bitcoins, or the smallest allowable bitcoin fractions, distinctly labeled entities? My understanding was that they are not, and that the blockchain only records information on the amount of bitcoins each user has. 

Also, if I am correct, what would double spending actually look like in the bitcoin system? If I send a transaction to one miner saying "I pay user X 1 bitcoin" and another transaction to a different miner saying "I pay user Y 1 bitcoin", then it seems like I haven't double-spent, but rather just paid two bitcoins in total, one to each user. Right?
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)