If Propionibacterium are obligate anaerobes (to wit, poisoned by oxygen), what makes 'Dairy' or classical propionibacteria (e.g. P. shermanii, P. jensenii, P. acidicpropionici, et al) viable in Fermentation starter cultures?
Here (mic.sgmjournals.org/content/127/1/121.full.pdf) is a link to a paper in which P. shermanii is grown in flask culture. Although the conditions are clearly designed to achieve low oxygen concentrations in the culture (the flask is almost full and is only shaken occasionally), they also indicate that oxygen is not toxic for this organism.
Also, this is from the WP page you linked to:
Obligate anaerobes are microorganisms that are killed by normal atmospheric concentrations of oxygen (21% O2). Oxygen tolerance varies between species, some capable of surviving in up to 8% oxygen, others losing viability unless the oxygen concentration is less than 0.5%.