you can't replace source of native vector layers in AE (Text and Shape layers) because indeed they have no source - they are generated in the timeline and not the project window. you can approach this in a number of ways - based on your screenshots the way I would do it is this:
1. drag the logo comp to the timeline and place it below the shape layer.
2. make the shape layer the Alpha Matte of the logo by clicking on the TrkMat Switch on the Logo Layer - Alpha Matte
3. Parent the logo to the Shape
I don't think track matte is the best option in this project. If there is any animation in the Square Box 1 layer then This is a better solution:
- Add the Logo (image or animated comp - more on your Logo comp later) to the LT15 composition just above the Square Box 1 layer
- Hold down the Shift key and parent the Logo to the Square Box 1 layer
- Turn off the visibility of the Square Box 1 layer
Holding down the shift key will snap the Logo to the same position and scale an rotation of the parent layer and pick up any animation you have in the layer in the same way Replacing footage would have worked if you had used a solid or an image for the Square Box placeholder.
I think you have some other workflow and procedure problems too. While it's true that using shape layers as placeholders is not a good idea there are some other inefficiencies in your project. This is the first thing that concerns me.
It looks to me like you are using a full sized composition with just a twitter logo in the middle as the source file for an element that is about 150 pixels square. I'm not sure what's in the Logo comp but if it's just a twitter logo then you would be better off using image (preferably vector) file. It will be a lot easier to work with. If you need a nested comp for a small element like that it is often better to make the comp the size of the asset. In this case if the place holder is 150 pixels square then the replacement comp should also be about 150 pixels square.
Another way to replace a shape layer with footage is to pre-compose the shape layer and then either put your replacement asset in the pre-comp or directly replace the pre-comp with footage. The only problem with doing that is that the any animation you added to the shape layer will not translate to the replacement footage.
It takes a while to get a good handle on how to use shape layers and to understand how shapes relate to their anchor points and shape transformations so you can use those values to control other layers.
If this were my project I would have used vector assets for the Logo layer and a solid for the placeholder instead of a comp. It's hard to say for sure because I don't have any idea what's going on in the nested comp or what kind of animation or effects you have added to the Square Box 1 layer.
I also noticed an expression error but if you can't get that solved it should probably be addressed in another post.