Could I have loose teeth due to menopause?

Tooth loss. Menopause does not mean tooth loss. Tooth loss is a function of poor oral hygiene, gum disease, bone loss (due to infection of the gum and not menopause or osteoporosis), tooth decay, accidents. So do a good home care, see your dentist for regular check ups and cleaning. And make sure you ask about the health of your gums.
Bone loss. Menopause increases the natural bone . Also you collagen in general deteriorates. Consider your past dental health. Keeping updated with your dentist is specially important to prevent further loss of these materials that make for healthy teeth and gums.