Can we put such applications in the Play Store?
What do you mean by such?
I think that is an all-in-one news app where the user can pick the website of his choice (from the list you see on the screenshot) and read news articles. Something like that I believe
Yes you can unless you will use ads in your app.
If you will use ads then Google will remove your app.
One of my app related to this is suspended when reach 20k+ installs. Due to the copyright content of news websites.