"To share a sensibility" means to have similar feelings, ideas, or values about something. It implies having a mutual understanding or connection with someone based on a common perception or outlook. Full definition
Her piece bears that out, sharing a sensibility with 1960s - era artistic experiments with shaped canvases. (washingtonpost.com)
If you feel frustrated that your wife has a habit of reading late into the night, you probably expected her to share your sensibilities for an early bedtime. (relevantmagazine.com)
But with few women in their middle years represented, and even fewer who share my sensibilities, I have found little real guidance from these sources in my middle years. (sharingajourney.com)