Someone says segment register is 16-bit in real mode and 32-bit in protected mode, is that right?

no ,is always 16 bits but 'mov eax,ds' is a valid instruction , bits 31-16 are set to 0

16bit indeed - however, there's a "shadow cache", invisible to you, that stores base/limit meaning the CPU doesn't have to consult the LDT/GDT every time a segment is used, only when loaded.

you can treat it as 16bit. In reality it is bigger because of speed optimization, but it beheaves like 16bit. (address of paragraph in real, index to descriptior table in protected mode)

Since the shadow cache is 100% invisible to programmers, whether it be ring3 or ring0 programmers, it "is" 16-bit. Imo it's somewhat weird that you can move a segment register to a 32bit register, since you're never going to get more than the 16 low bits.