Is any of my personal information still contained on a defunct motherboard that I have had replaced in my computer?

On older motherboards, the board itself should contain almost zero personal information. Be sure to remove any hard drives or solid state drives, including newer types of storage like NMVe or M.2 devices mounted into the expansion cards. If you used a BIOS-based system password (that appears as soon as you power on the machine, before Windows boots), this will be stored on the flash ROM, which is often removable. Newer boards have a Trusted Platform Module (TPM), which contains a private key, which might have been used to encrypt your drives or verify software. Some TPMs are removable. If there's no TPM or BIOS password, your board should not contain any personal information as far as I know.