I have parent object, which I rotate left/right on Y axes using animation.
And inside it I have child object, which I want to rotate up/down with another animation.
For this I want to create 2 different animations, but when I try to create animation for child obj, it doesn’t create new animation, it opens animation of the parent object. I tried to create new clip & make animation for child obj, but it just doesn’t move the object like there is no animation at all…
How can I create 2 different independent animations?

In your animation window and with your parent selected, click on Add Property, and then find your child’s name in the list and add its rotation.