Your custom account is not related with the app pool identity. App Pool Identity something that belongs to that IIS app.

For the custom account it changes.

1)IIS and SQL can be on same domain/network (generally dont but as a scenario i need to explain it) If yes, on SQL server side that login should be created or should be inside a group which is also defined is SQL Server as a login. So your login probably inside a group which has powerful authorizations which can be dangerous actually. If IIS ve SQL are on same domain/network they can be on DMZ network which is critical. Or maybe you open some private network to outside.

2)IIS can SQL can be on different networks -> This is generally the normal case and if this is the case the group has also capable of ask the necessary LDAP to get auth. This is also dangerous at least two networks should be protected in this case.

So in my opinion, you should use app pool login or you should create a sql login to manange these kind of structures. Otherwise you can not be able to monitor what is going on at sql server and this is dangerous. It is also more easier on prepating connection string also.