Is there a way to change skin tone on family emojis?

There are no families that are the default orange color, either. Not unless they're suffering from severe jaundice.

Many of the old group emojis are no longer part of the set. These are the only ones I could find that depict more than one person.

And all of them can be tapped/clicked on to choose a different skin color.

You are free to submit feedback here:

Product Feedback - Apple

Not sure which icon you are referring to, but you can use your actual Memoji to identify each family member:

Other icons are typically blue and white with no significance toward skin tone. It is simply an icon. Difference shades of blue and purple are also used without implying any skin color. Notice that the above pic shows an icon to add a person that is blue with a white background.