Is it now possible to spy on other person's iPhone using microphone? There is this app on the Appstore "Find My Kid" and "Chat with parents". They allow to the first app to start listening to microphone of the second app even without notification. Even when the app is in the background and the screen is off. 

So I wonder: How does the app do this?
I think the incorrect part of your question is the word 'spy'.  No, it's not possible to spy or listen to an iPhone's microphone without a notification or without the users consent.

I'm not a developer of these apps but I am an iOS developer with a few apps in the App Store so I'm familiar with the iPhone's API's and App Store review process.  Apple does not provide a way to secretly listen to another device

From reading their product page description, the 'listening' is done using a chat application or Messages.  The app starts recording and then sends the recording to the parents via chat.  It might work different in Android but for iOS that's what it looks like it's doing.  The image below specifically calls out "Android/GPS watch".



There doesn't appear to be anything 'secret' or in violation of Apple's app guidelines in these apps.  The wording on the app's App Store description is vague regarding how they do what they claim.