That bitcoin is done like this:

---alices signature---
coin abc
---bobs public key---
---alices signature---

instead of like this

---alices signature---
alice: 0 bob: 1
timestamp (to prevent replay attacks)
---alices signature---

where bob and alice are replaced with keys, and the network just keeps a running track of who has how many points (and if alice says to give bob more coins that she has it is rejected by the network, and if alice and bob do not equal out to a possible difference it is rejects (for example alice has 2 coins and bob has 1 coin prior to a transaction, so alice: 0 bob:4 will be rejected but alice:1 bob:2 and alice:0 bob:3 will not be). Wouldn't that make the network more anonymous, as 1 coin of Alices would be equal to one coin of Bobs rather than there being a difference between coin abc and coin def?

Is there something I am missing?