Tying together
The bitcoin network does not really tie bitcoins together to form UTXOs.
The only sense in which it does this is in transaction data. The details of a transaction includes a list of input UTXO's which are being spent/consumed/destroyed and a list out output UTXOs which are being created/received.
Every full participant in the Bitcoin network has their own journal of all transactions since the start of Bitcoin. This is used to independently check validity of received data and especially of UTXO validity.
You might say that a transaction ties together input UTXOs to output UTXOs but it doesn't tie them individually. It doesn't tie one input to one output if there is more than one input and more than one output.
Keeping track of money
The Bitcoin network doesn't keep track of people's balances probably because there didn't seem to be a reliable way to do that in a peer-to-peer network that is both designed to eliminate any need for trusted third parties and to promote privacy. You probably don't want everyone in the world to know exactly how much money you have and exactly what you spend it on.
Coins
Bitcoin is a unit of measure like seconds. You can't hand someone a second or lock a second in a safe. You can't really point to something tangible and claim it is a second.
The nearest thing to a coin is a UTXO but you must remember that the idea of a coin is an analogy, it isn't real. It is a story told to help understanding, but stories may also mislead.
You might point to a dollar coin and say that is a dollar but really it's a token, it isn't intrinsically worth a dollar. Most money you have ever used in your life isn't really real. Numbers in a bank account are even more abstract.
Details of an amount of Bitcoin money is stored in the transaction journal we call the blockchain. By inspecting this we can see that amounts of money have been spent and received. We can see by inspection that those spending money have proved they have the right to do so. They proved they know a secret number associated with an amount of money, without having to divulge that secret number.
So no, a Bitcoin coin is not literally a string of code that can be copied or divided.
Other stories
The world of Bitcoin uses many analogies that are potentially misleading and lack of knowledge of the truth can sometimes cause people new to bitcoin to make mistakes that harm them financially.
Wallets: Wallets are not wallets. This is because they don't contain money. Wallets are really a kind of key-safe used to hide a secret number we call a private-key.
Addresses: Bitcoin addresses are not an address. They don't identify or locate people, organisations or places. Just knowing an address or just having it in a wallet doesn't give you any ability to spend money sent to that address. An address is perhaps more like a purchase reference.
Ledger: The blockchain is not a ledger, it is a transaction journal.
Keys: A Bitcoin key is a big number resulting from a mathematical calculation. Bitcoin keys don't really open doors or safes or do anything really similar to that. They are used in the creation and checking of digital signatures. The type of keys known as private keys can give you the ability to spend money indirectly associated with that key. The number known as a private key can be used in a mathematical operation to produce another number I have called a purchase-reference (address). You can also use the private-key in a different mathematical operation to produce a number that the rest of the world will accept as proof that you may spend money associated with that purchase-reference (sent to the address).
Signatures: Digital signatures are not like written signatures. They are not a sequence of symbols or squiggles that can be copied and attached to a new document.
Encryption: The Bitcoin network doesn't use encryption
Coins: There aren't really any coins in the Bitcoin network. UTXOs are a little like coins but weird ones that, every time you spend them, get melted down together and re-cast into a different set of new ones each with different values. A coin is just data about a UTXO but you never have exclusive custody of any coin (UTXO data). A single coin (UTXO data) is simultaneously everywhere in the Bitcoin network and nowhere else (not in an exchange or trading account for example).
Tracing: Although every movement of money is recorded in a public record that everyone has a copy of, you can't tell which inputs to a transaction correspond to specific outputs. A transaction might have inputs of 7 and 3 but have outputs of 9 and 1. You cant tell whether the 1 came from the 7 or from the 3. This makes it hard to track money.
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