I signed up for coinbase and i have to sign this agreement which i have to check. It also ask me to write my legal name. I obviously didn't think much about this and would put my real name but then i read people wrote a fake name there? If so, isn't this going to be a huge problem since they verify your name and bank account?
You have to use your real identity to purchase through Coinbase.
If you want to remain anonymous, can use localbitcoins, but it will be more expensive.