Assemblers can come from the CPU manufacturer, but they can also come from a third party. They're most often bundled into the compiler, since it's rare to need to compile without also assembling (many compilers skip the assembly step and convert directly to machine code). There's actually a third tool involved I didn't mention, which is a Linker (takes pieces of machine code and combines them together, so you can separate your software into different parts).

Some CPUs are more niche, and the assemblers/compilers for them sometimes are created by the manufacturer. But there are a few "standard" processor architectures that many CPUs follow, and because they're common between many brands of CPU (and the architectures are published and well known), the compilers are often created by third parties. And there are some compilers (like gcc) that can compile to dozens of different architectures.

As an example, you might have heard terms like x86, x86_64, armv7, or RISC-V -- these are all processor "architectures", which means that any processor that meets a particular architecture can run machine code built for that architecture. Currently, x86/64 architectures are common among desktops and most laptops (Intel and AMD processors), while ARM architectures are common for tablets and phones and some laptops (Snapdragon processors, for example).