"Ok, you might say 'Assembler',..."

I absolutely would!

"but I see it as compiler, too ... sort of ..."

Absolutely not!

The tool which converts assembly-language source into object is called an Assembler;
The tool which converts high-level language (HLL) source (eg, 'C') into object is called a Compiler.

Assemblers and Compilers are two very different animals, and not to be confused; the terms are not interchangeable.

See my earlier discourse on the subject:
http://www.8052.com/forum/read.phtml?id=28970