I don't have twins but I totally get where you're coming from with wanting people to see them as individuals. I'm sure you're already doing what you can to foster their own identities, hobbies, friends etc. so other people's behaviour won't have as much of an impact on that. I think that making sure they have different hairstyles and don't dress the same etc can take away some of that stigma that twins can suffer when people can't tell them apart.
I do think people would just think you were being awkward if you asked them not to refer to them as the twins, but its worth making a point of not doing that yourself.
My DP has two girls (not twins) and always refers to them as "My girls" which I find odd, as I have boys and girls and just call them all the kids. I feel like highlighting that they're girls every time you talk about them makes their sex more of a thing than it needs to be. Similarly saying "the twins" is bound to highlight their sameness over and above their individual personalities.
My DCs have had several friends who are twins and I always thought it was interesting when we would invite one of them to play (the one that DD actually liked spending time with!) and their parent would be so thankful that we'd only invited one, not both of them as a pair.
I guess lots of parents thought it would be mean to only invite one, but to me it seemed strange that my DD couldn't just spend time with the one she liked without her DSis being there too! I'm sure if you make a point of promoting separate friend groups, allowing them their individual style etc they'll find their own way to deal with the twin thing with as little competition as possible!
I find that twins who try to look alike even as adults have that competitive thing going on a lot - Darcy and Stacey from 90 Day Fiancé and Michelle and Sharon from Married At First Sight Australia spring to mind! Maybe TV reality shows aren't the most likely place to find healthy sibling relationships anyway, but these sisters were both eaten up by rivalry and jealousy! In both cases even as adults, they had the same hairstyles, the same cosmetic procedures etc to try and remain as similar as possible.
There must be a constant push and pull to try and maintain that feeling that they're 'special' due to the twin thing, but also to try and find their own place.