Can you classify risks based on the Project Objective a risk would impact?

Yes. we can also classify risks based on the Project Objective a risk would impact. They are:

a. Scope Risks - Risks that are related to changes to the Project Scope (Ex: Scope Creep)

b. Quality Risks - Risks that are related to the Projects Quality Standards (Ex: Missing Quality checks)

c. Schedule Risks - Risks that are related to the Projects Schedule (Ex: Missed Delivery dates)

d. Cost Risks - Risks that are related to the Projects cost (Ex: Budget Overruns)