I completely agree with M0nica. Those 1980s floral skirts etc are so ageing. I think I dress younger now than in the 1980s - sheesh, I even had an old lady perm then!
I hit 60 last year and, apart from no longer wearing jeans which finish somewhere south of my belly button, I dress as I like and, yes, some items are the same as my daughters-in-law wear. If you feel confident and comfortable (hence the above the belly button jeans) then wear it. If you feel uncomfortable or inappropriately dressed, it'll show on your face and that's certainly ageing.
Oh the only other concession I have made is to make sure my feet are comfy - again shows on your face if you're in pain!
Other than what I've said, and which may only apply to me, there's no need to change anything just because you're about to hit 60 - at least I don't think so and no-one actually laughs at me, at least not when I can see ...
If you think you should change your appearance, why not investigate a personal dresser in somewhere like Debenhams and do it slowly, maybe buy a few staples like a 'good' dress or trousers.
PS It's ok being 60 ... not sure about 61 though, which is what I will be this time next week. Still not buying floral skirts or lavender cardies though.