This error has had me running round all the houses today, it was so annoying.
You need to set the Identity that IIS is using to that of Network Service.
Many thanks to zillionics. for pointing out that not only does your webapp have to have it’s app pool Identity set to Network Service but so does the DefaultAppPool (I guess there was a clue there in the error message!)
Here’s a link to their post: login-failed-for-user-iis-apppooldefaultapppool
In case you are having MSSQLServer issues here is a link on how to set the the permissions to allow the Network Service have access to your database: