Does the installation of tiko have any influence on the contract I have with the utility company?

No, the installation of tiko has no impact on your contract with your utility. It means that you will still receive and sell electricity to your utility without any change in pricing. You will just save on your bill!

How does my registration to tiko storage change in my relationship to my energy provider and my tariff?

Your registration to tiko storage is completely independent from your relationship with your electricity provider. Your tariff won't change.