I can’t off the top of my head think of a way that would be easier than using a different layer for each circle.
You could use the magic wand to select different circles and edit them individually, but I don’t think that’s really any simpler than just using different layers to begin with.
Honestly, Photoshop is the wrong tool for this. You can do a hugely overlapping number of things in Photoshop, Illustrator and InDesign, but each has areas where it can do tasks more easily and with less effort than the others. (And in concert - they can do anything at all.)
If you have Illustrator and any skill with it, you’d be better off working on this design in that. Otherwise, the best suggestion - layers for each object - has already been made. The one thing PS is worst at is scalable, redefinable objects. Even its shape tools are kiddie versions of what Illo does.
In Illustrator, you can draw a circle, and define it to be a ‘symbol’, Adobe’s name for a master copy of something. You then place ‘instances’ (copies) of the symbol in your drawing. They can be different sizes and orientations. If you edit the symbol (changing the colour, for example), all the instances change too. On the other hand, you can edit an instance and break its dependency on the symbol, creating an independent object.
A ‘vector’ drawing program like Illustrator is better for doing things like logos that will need to be resized and scaled. They create artwork that is ‘smooth’, and, unlike pixel-based images, can be resized without losing accuracy.
You can certainly import an Illustrator object into Photoshop and ‘rasterize’ it (convert it to an image made of pixels). But this is best done as a last step, to make a pixel-based image that is the exact size you need it to be. Resizing pixel-based images, especially non-photographic ones like logos, often leads to poor results unless you know exactly what you’re doing.
I often create separate files for a complicate design. It’s easier to deal with a small file than one big file with 25 layers. The OP mentions various sizes and colors. One of those would be one file. Resize it like you want. Set the background color with the Fill command. Add your circles. Do the next File.
Then start assembling. Flatten layers (small file), then ctrl-c to copy from your small file, ctrl-v to paste on your logo.
It’s not necessary to waste time saving those small files. Create them, use them, copy/paste and discard. Unless this is something you’ll be working on for several days.
I do a lot of photo collages. Each photo is a separate file. I edit it, add a levels layer, hue adj layer. Flatten and copy/paste to my big file. Works really well.
Photoshop has supported basic vector shapes for years.
If you don’t have or want to use Illustrator, just use the Photoshop shapes tool to create easily-modifiable vector shapes. You can layer, group, or intersect/combine/subtract/etc. them as you like.