I'm kinda sympathetic to this piece of writing, without completely agreeing with it.
I wouldn't call mothering a job. I think in the quest to make mothering seem valuable, people call it a job and put a monetary value on it.
But it's not at all on the same level as paid employment. You're your own boss. There's no bottom line, no co-workers, no company, no "business".
Your job is to feed, clothe,clean and educate a little being every day. That's not a "job".
I wouldn't call it a privilege, either. It's not something special granted to you. You can do it, or not. Just because you get to do it and some people don't doesn't make it a privilege.
And I definitely wouldn't call it a hobby. A hobby is an interest. Mothering is far above a mere interest that you can leave behind at will.
That said, I do agree that calling it a "job" is ignorant and condescending, as if mothering doesn't have value unless it's on the same level as going to a job. I would call it a vocation. It's an important task you devote yourself to, that isn't done for the money or any accolades. It's just important in itself.
I think that parenting is the most difficult "job" in the world (for lack of a better word). On the other hand, you don't have to remind everyone of your martyrdom. On balance, it's probably easier to raise kids yourself than go to a job 8 hours a day and then come home to domestic cares. It's definitely nice not to have a job to worry about.