Fmri doesn’t measure blood flow, it measures the oxygen level in the blood. Hemoglobin molecules change shape when they carry oxygen and the different shapes react differently to magnets, which is a real stroke of luck
it doesn't measure the oxygen level directly either. the bold signal is correlated to dephasing induced by the oxy/deoxy hg ratio that isn't even necessarially localized to the voxel (flow or long range magnetic susceptibility perturbations from nearby accumulated deoxyhg (veins)).
Yep, this is why it's also called BOLD imaging, for blood-oxygenation-level-dependent fMRI. I did my PhD is BME and brain-computer interfaces, but it has been a while since I worked in the field.