Modern solid-wood furniture makes use of lighter materials too. Sometimes the backboard will be thin particle board, or certain planks will be pretty thin. Or the drawers are made of much lighter wood, possibly veneered. Also, the designs are much less ornate.
So in general, your modern solid wood furniture AND cheap, flatpack stuff will be lighter that old well-built furniture.
This is true—an IKEA Billy, say, is really heavy for the volume of material involved (compared with the same amount of oak, for example) but is designed for cheap + packs-flat, not to minimize weight. It does win on size and stackability, when disassembled.
Ehhhh… sometimes.
Solid wood’s usually way lighter than the same thing built in particle board or plywood, though.