Indeed xchg eax,eax is a nop idiom; there are many. Recent microarchitectures simply ignore nops without executing anything. From the Intel® 64 and IA-32 Architectures Optimization Reference Manual:
16.2.2.6 NOP Idioms
NOP instruction is often used for padding or alignment purposes. The Goldmont and later microarchitecture has hardware support for NOP handling by marking the NOP as completed without allocating it into the reservation station. This saves execution resources and bandwidth. Retirement resource is still needed for the eliminated NOP.
This nop idiom is very special however, since it isn't just about efficiency: if it wasn't a nop idiom, xchg eax, eax would not be a nop at all, it would clear the upper 32 bits, as xchg ebx, ebx does (or any other register other than eax).
16.2.2.6 NOP Idioms
NOP instruction is often used for padding or alignment purposes. The Goldmont and later microarchitecture has hardware support for NOP handling by marking the NOP as completed without allocating it into the reservation station. This saves execution resources and bandwidth. Retirement resource is still needed for the eliminated NOP.