I have only been asked for my drivers’ licence when I will be using their vehicle, and to me that’s understandable.
My guess on the Facebook request is that the homeowner is hoping to get a glimpse of ‘the real you’. No doubt we all are on our best behaviour when applying for a sit, including any video chat, just as we would if it was a first date. Reviewing your social media may allow them to see you in a different light.
I have no concerns about sharing my social media links, as well as my blog. If it helps the homeowner to decide that I’m a good fit (or not), I’m fine with that.