I’d consider a sample size of 170 to be pretty large, if the sample was drawn with perfect randomness from the population. But this one wasn’t, it was self-selected. Also wasn’t a clinical trial, and while they seem to know what they’re doing with setting up the questionnaire, I would assume it would result in larger measurement error, which would need more samples to be able to correct for.
Completely agree with you though - the conclusions that it seems reasonable to draw from this are ‘not much, really’. Seems to disagree with the results of a larger study by many of the same authors, too, which say that companion animals did result in a smaller decline in mental health during lockdown.