Love geometric animation!
There may not be a 1-step answer, but I can see some ways of approaching a design problem like this. When working with geometric shapes for animation, sometimes it can help to break up a polygon into its smallest repeateable parts, and rebuild your shape in AE, rather than using just the shape tools available to build a whole shape (although always a great place to start).
You can get a lot of milage out of adding a repeater to a shape layer for example (toggle open the shape later, go to the add dropdown and select repeater). Since a geomtric shape like this radially repeats, you can design one segment of a shape, and then set up your repeater to settings that reduild the shape you want. Then, you only have to animate one segment of the shape, and all that animation will be reflected in the other instances of the shape. The main limitation of this is that you can't offset the animation at all - all the repeaters are just copies of the shape, so they all animate at the same time. BUT this is pretty easy to use since you only have 1 set of keyframes.
You could also manually copy and rotate animated segments in multiple layers to give you more control over offsetting the individual segments in time.
Also you can convert any parametric shape into a bezier path by right clicking the "path" group and selecting "convert to bezier path." That makes all the points available to you, but you will have to individially keyframe each.
Oh, there's too much going on in this post already, but on a shape layer going to "add > trim paths" is also the best friend of many a motion designer. It gives you the ability to do a "draw on" effect on a shape (looks best on strokes). So you can look into "trim paths" to help make your shapes draw on and off. It's a useful tool for morphing because you can draw a complex set of shapes that are all connected on one layer, but only show a small section of the shape at a time by animating "start," "end," and "offset". Not great for this exact example, but check it out for future reference.
Adding a few screenshots of the repeater method! It rebuilds just the outer edges of the shapes.
I'm making some assumptions about your final look, but if you want to get something like the AI blend effect (that animates) you can play with the effect time --> echo after you have animation on your layer. That's the last screenshot.
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