No the iPhone 5s battery is not compatible with the iPhone 5.
The battery connectors are not the same.
In addition, the iPhone 5 battery spec is: 1440 mAh, 3.8V, 5.45 watt hours. The iPhone 5s battery spec is: 1558 mAh, 3.8V, 5.92 watt hours.
Don’t try and use an iPhone 5s battery in an iPhone 5!