Some babies like to play with your jewelry or hold onto a blanket while they nurse. For other babies, however, the toy of choice is your nipple. It seems awkward, uncomfortable, and maybe even funny, but believe it or not, many moms are asking "why does baby grab my other nipple while breastfeeding" out of frustration, annoyance, or (sometimes) with humor. Whether or not your baby plays with your other breast only on occasion or can't seem to eat without a "firm grip," you've probably asked "why" a time or two.
According to La Leche League International (LLLI,) it's pretty common for babies to fidget while they nurse. The reason why boils down to personality — some do and some don't. Breastfeeding Basics noted that the behavior is seen in most mammals as an instinct to encourage letdown, similar to kneading the breast.
According to the Dr. Sears website, "twiddling" is most common in babies 6 to 9 months old, and the habit usually dies down or becomes less of an attachment at that point. If you plan to nurse past infancy, many toddlers pick the habit back up again, as they're naturally prone to move and fidget.
Regardless, this little habit, whether instinctual or not, can be a bit annoying and even cause you to resent nursing if you feel like you can't tell them to stop grabbing. However, you don't necessarily have to put up with something that bothers (tickles/amuses/annoys) you if you think it will get in the way of nursing effectively.
If it doesn't let up on it's own, you can try finding substitutes for your baby to hold onto instead. Introduce a nursing necklace, make your breast harder to access, give them another toy or blanket to hold onto — the options are limitless, and what is most effective will be different for each baby.
If you have a "twiddler," as the internet so fondly puts it, don't despair. The behavior isn't permanent and most moms are able to replace it with a different, less invasive, action.