Sounds like this guys is trying to make it so each of your own insurance company covers your own cars, which is bullshit if it is his own fault. Did he say if his insurance company is going to be contacting you? I would still wait for them to call you. Tell them that he admitted fault… you should have asked the guy… did you make sure you told them it was your fault?
If you go through your insurance company, it will be a mess, you will have to pay your deductible, it will count on your policy, and you will have to pay for a rental car if you don’t have it separately on your policy, which is all bullshit… just wait to hear from his insurance company… when all else fails, then call yours.
I have been through this a million times… you always have the option to make a claim with your insurance company down the road if he jacks you off.