Congratulations on your new iPhone 5s, and thank you for the details of the issue you are experiencing when connecting your iPhone 5s with your Suzuki Kizashi's stereo.
In my experience, when updates are released for iOS devices, compatible devices release an update to maintain compatibility. Step 2 from the article below recommends checking with the manufacturer of your car stereo to see if there is a firmware update available for your car stereo:
2. Update software and firmware if necessary
Verify that both your car stereo and your device have the latest software versions. Contact the manufacturer to see if any updates are available. If your device does not have the latest version of the iOS, update to the latest software version.
I recommend reviewing the full article here for additional troubleshooting steps:
Thank you for using Apple Support Communities.