This "sharing" is daily? Oh my goodness. And what exactly does "expressing" mean? "Sharing," just the word bothers me. I'm not "sharing" anything; I'm telling you something. The use of the word is enough to set my teeth on edge.
In the past, I've found that "sharing" too much of my personal life at work has had negative consequences. Not forced sharing, as in the OP, but just talking about my private life.
Talk about sewing or crocheting or bring in cookies I've baked? I'm Little Miss Homemaker, and therefore not management material, or even deserving of a promotion, because my heart isn't in my job; it's at home.
Talk about nephews and nieces? Then I'm a homebody and wouldn't want a job that involved any amount of travel. Even though I'm single and childless and you'd think I'd be a perfect candidate for traveling.
Mention that I like science fiction? Then I'm weird. Mention that I like medieval history and literature and even read the literature in the original Middle English? Weirder still. Comment that I don't watch Survivor or American Idol or really any reality show? Seriously, I've been called un-American for that, and I'm not sure the person was joking.
Add in my personal reluctance to talk about any gentleman I might be seeing, and the fact that I attend office Christmas parties without a date, and that I have cats and talk about them, and I'm the crazy cat lady.
Then add in having to come up with something to "share" every single day? Good grief.
In the OP's shoes, I'd trade on my love of history and language, and have an interesting word or historical fact to share. If challenged, I'd simply state that these are my hobbies, and I thought that's what we were supposed to be "sharing." If pressured to "share" more personal thoughts/feelings/happenings, I would somehow try to make it clear that I feel *very* uncomfortable about bringing my personal life into work. And question why my supervisor felt it was necessary for me to do so.