imul (signed) and mul (unsigned) both have a one-operand form that does edx:eax = eax * src. i.e. a 32x32b => 64b full multiply (or 64x64b => 128b).
186 added an imul dest(reg), src(reg/mem), immediate form, and 386 added an imul r32, r/m32 form, both of which which only compute the lower half of the result. (According to NASM’s appendix B, see also the x86 tag wiki)
When multiplying two 32-bit values, the least significant 32 bits of the result are the same, whether you consider the values to be signed or unsigned. In other words, the difference between a signed and an unsigned multiply becomes apparent only if you look at the “upper” half of the result, which one-operand imul/mul puts in edx and two or three operand imul puts nowhere. Thus, the multi-operand forms of imul can be used on signed and unsigned values, and there was no need for Intel to add new forms of mul as well. (They could have made multi-operand mul a synonym for imul, but that would make disassembly output not match the source.)
In C, results of arithmetic operations have the same type as the operands (after integer promotion for narrow integer types). If you multiply two int together, you get an int, not a long long: the “upper half” is not retained. Hence, the C compiler only needs what imul provides, and since imul is easier to use than mul, the C compiler uses imul to avoid needing mov instructions to get data into / out of eax.
As a second step, since C compilers use the multiple-operand form of imul a lot, Intel and AMD invest effort into making it as fast as possible. It only writes one output register, not e/rdx:e/rax, so it was possible for CPUs to optimize it more easily than the one-operand form. This makes imul even more attractive.
The one-operand form of mul/imul is useful when implementing big number arithmetic. In C, in 32-bit mode, you should get some mul invocations by multiplying unsigned long long values together. But, depending on the compiler and OS, those mul opcodes may be hidden in some dedicated function, so you will not necessarily see them. In 64-bit mode, long long has only 64 bits, not 128, and the compiler will simply use imul.