Units of measure
Satoshis and Bitcoin (BTC) are mainly a unit of measure. 100,000,000 Satoshi is the same magnitude as 1 Bitcoin just as 100 US cents is the same magnitude as 1 US Dollar.
A bitcoin wallet or other software might display the amount in Bitcoin or in some other units instead of in Satoshi. This is just a choice of presentation. Just like you might write down 5 cents as $0.05 or as ¥5.7 or €0.043 (at today's exchange rate). Writing it down any of these ways doesn't change the nature of the actual 5 cent coin in your pocket.
The Bitcoin network keeps track of money as amounts using only Satoshi as it's unit of measure. The list of "coins" is a list of unspent transaction outputs (UTXOs) derived by processing the list of Bitcoin transactions that acts as a transaction journal and which is replicated by each bitcoin node.
when I have ownership of 100,000,000 Sats I can say I have the equivalent of a full bitcoin?
Yes, just as when your pocket contains five US quarters, a nickel and a dime, you can say you have one dollar forty. A similar approach can be taken for Bitcoin amounts or for any currency really. You can express a total without enumerating the coins.
ID or serial numbers for coins?
is it actually that I own sat number 241536 from BitCoin number 237518 or whatever?
The coins are identified by the hash of the transaction that created them and their position in the list of outputs of that transaction.
However, Transactions destroy coins. Every transaction effectively melts down all coins that are input to the transaction, stirs the molten metal thoroughly and then casts a set of completely new coins for the outputs of transactions.
Note that Transactions don't link old coins to new ones individually. All input coins are destroyed and do not have any individual link to specific newly created coins.