How to verify installation of a new battery in an iPhone?
Is there any way to confirm that the battery actually got replaced?

There are a couple ways you can use to check the battery status/health on your iPhone:

Check for your iPhone's battery's maximum capacity under Battery Health status in the iOS Settings app.

https://support.apple.com/kb/HT208387


Maximum battery capacity measures the device battery capacity relative to when it was new. A battery will have lower capacity as the battery chemically ages which may result in fewer hours of usage between charges. Depending upon the length of time between when the iPhone was made and when it is activated, your battery capacity may show as slightly less than 100%.


A newer/recent battery should show a number close to 100% for the Maximum Capacity value. You can access this information under Settings app → Battery → Battery Health on your iPhone.

Check the battery charge cycle count and charge retention capacity on your iPhone.
You can check the battery charge cycle count and charge retention capacity on your iPhone by using a 3rd party app called coconutBattery. A charge cycle count gives an estimation of full charge and discharge cycles endured by your iPhone.