Yes, TWRP is a “clone” tool. I use it when i want to test ie sailfish, and then return to my everyday OS. Very useful, since other solution perform incomplete backups.
To my knowledge, you can do a “fastboot flash recovery recovery.img”. Are you sure you took the correct TWRP recovery image ? In case of doubt, you can try the community compiled version here : ✏ Porting TWRP recovery
If for any reason you don’t want to flash, only to run it, then you’re supposed to use “fastboot boot …”. But this one doesn’t work, it requires the image to be signed. @retsifp found out how to perform this signing, see here : https://forum.fairphone.com/t/fastboot-boot-is-broken/23235/2