I'm not so worried about the "my grandchild" part. I think most grandparents refer to their children's children that way. My MiL refers to my daughter as her grandchild all the time, and I hear my mom refer to the other grands that way (my kid is just Name when we talk). It's not possessive. It's just a natural tendency (in English, at least) to identify a person in terms of their relationship to yourself, the same way I used "my MiL" up there and not "my spouse's mother" or "my daughter's grandmother."