Is it possible to change layer select like how it is in illustrator?

It’s the Auto-Select option that Melissa Piccone mentioned. I wanted to add that there are times when it might select a different layer than you think you’re clicking on, depending on the opacity of their pixels. This is not a bug, selecting is naturally ambiguous when pixels aren‘t 100% opaque. What you can do then is right-click, which will list all layers under the pointer, and then you can choose the one you really want.

I'm finding Photoshop to be insanely user-unfriendly and lacking a lot of logic that illustrator and Indesign do have.

By @JimmyLChannel

Many would find that to be a very very odd observation. If Photoshop seems unfriendly compared to Illustrator and InDesign, that‘s probably because Photoshop is designed for pixel graphics, which is very different than the vector graphics that Illustrator and InDesign are built for. The reason you’re thinking they’re not designed well is that your expectations have been set by working with vector-based applications. But if you understand both sides, you wouldn’t design them in exactly the same way any more than you would design a screwdriver to work exactly like a hammer, and then try to drive screws with that…

Another way to put that. If you’re convinced that Photoshop was designed badly, the problem you are going to have is that every other photo editing app you try by other companies will also seem designed badly, because they all kind of work the same way…they’re all pixel editors, not vector editors. It would be like being an expert race car driver, getting into an airplane, and telling the pilot that the airplane controls are “user-unfriendly”’ and “lack logic.” That’s not the reality, the reality is that they are designed for their specific purposes, and for those with the appropriate training, both sets of controls make sense.

Auto Select allows you to select objects on any layer. It has it's limitations - if you have a transparent gradient or object on a layer above it will select that and using clipping masks can be a little tricky. But mostly it works as you are requesting. 

Select the Move tool, change the options in the Options Bar.
Like ID and IL, you need to spend a little time and learn how to use the apps.