Nowadays some cards are formatted with the exFAT filesystem when you buy them. TWRP seemingly can’t handle that currently (but Android can, that’s why it often goes unnoticed).
Backup other important data from the card if there is any, and then use your computer to format the card with the FAT32 filesystem, if possible, and try again.
… means if your computer is prepared for it (e.g. Windows installs drivers for this when you connect the phone to it for the first time), you should be able to transfer files with your computer’s file explorer after booting TWRP.
MTP is just the term for the technology behind it, and TWRP is able to do it.