What is the name of this effect present on the rectangle (e.g. light brown rectangle), and how is it done? The rectangle does not have a constant border radius, and gives a 'fun'/ friendly feeling.
I took it from Dribble, available here.
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I don't think there's a specific name for this kind of shape, however you could do something similar with Bézier curves in almost any vector software.
Start with a normal rounded rectangle (i.e. rounded by radius), and pull the Bézier handles out horizontally and vertically, perhaps using a grid to make sure it's exact, and repeat for each corner.
This example was made in Inkscape, but the same is possible in Illustrator and other vector software.