The health and comfort of the sleeping person - that's what really stands behind this belief. When someone's asleep, they're in a vulnerable state, and a bright camera flash can scare them, wake them up, disrupt their sleep. Back when there was no electricity and people slept in darkness, any unexpected light was perceived as a threat. That's where the fear came from - not something mystical, but a pretty practical concern that over time turned into a superstition about some otherworldly consequences.
Your cat is a completely different story. Animals are sensitive to light and sound, but they don't believe in omens, and if your cat is sleeping peacefully, it means the flash either didn't wake it up or it's already used to the phone nearby. The only problem would be if you're constantly waking your pet with flash - that can cause stress over time and mess with its sleep schedule. But a video of a sleeping cat by itself won't cause any harm.
By the way, what shooting mode did you use - automatic flash or night mode without light? If you've already shot a video with flash and your cat was sleeping fine through it, then there's really nothing to worry about.