Yes. Here's what happens when you already have Apple One through Apple or another third party. When you change the associated phone line to Unlimited Ultimate, Unlimited Plus or Unlimited Welcome* and choose the Apple Music Family perk, the Apple Music Family plan through Verizon and the original Apple One plan run at the same time.
You can cancel the original Apple One subscription directly with Apple or the third party.
Note: You'll be able to enroll in the perk with the same Apple ID to keep your original Apple Music account.
*If you have Welcome Unlimited you can also add perks in My Verizon. Welcome Unlimited is no longer available to add to your account.