Android is as Linux as every other Distro is.
It uses the Linux kernel
Yes because that's actually what Linux is. A kernel. There are also no such things as "Linux applications". Different distros use different package formats. If you mean ELF binaries you can certainly run those. Android in it's typicall configurations tried to prevent that because it would be a huge security vulnerability if apps could just run random ELF binaries, but if you root your device you can run those.
The names are pretty clear and are about form factor. Desktop is something on top of a desk. Laptop is something on top of your lap. Hand-held is something you hold in your hand.
The steam deck is a hand-held game console - doesn't matter what OS is it uses. It's true that most stat tracking sites count it as "desktop" but not because it's a desktop computer but because the user agent looks similar to desktop user agents.
If I install Android on a tower PC it doesn't randomly become a smartphone even though all browser trackers would register it as a smartphone.
And Valve using a "typical desktop OS" on their handheld console doesn't magically turn it into a desktop PC.