Scala, 92 bytes

def^(s:String)="(.)\\1*".r.findAllIn(s).map(_.size).reduce((x,y)=>(BigInt(x) gcd y).toInt)>1

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PS. Original solution was based on split function but it's longer (95 bytes).

def^(s:String)=(s+s).split("(.)(?!\\1)").map(_.size+1).reduce((x,y)=>(BigInt(x) gcd y).toInt)>1

Scala, 80 bytes

s=>"(.)\\1*".r.findAllIn(s).map(_.size).reduce((x,y)=>(BigInt(x) gcd y).toInt)>1

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PS. Original solution was based on split function but it's longer (83 bytes).

s=>(s+s).split("(.)(?!\\1)").map(_.size+1).reduce((x,y)=>(BigInt(x) gcd y).toInt)>1

Scala, 72 bytes

def^(s:String)=(s+s).split("""(.)(?!\1)""").map(_.size).distinct.size==1

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Scala, 92 bytes

def^(s:String)="(.)\\1*".r.findAllIn(s).map(_.size).reduce((x,y)=>(BigInt(x) gcd y).toInt)>1

Try it online!

PS. Original solution was based on split function but it's longer (95 bytes).

def^(s:String)=(s+s).split("(.)(?!\\1)").map(_.size+1).reduce((x,y)=>(BigInt(x) gcd y).toInt)>1

Scala, 68 bytes

def^(s:String)=s.split("""(.)(?!\1)""").map(_.size).distinct.size==1

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Scala, 72 bytes

def^(s:String)=(s+s).split("""(.)(?!\1)""").map(_.size).distinct.size==1

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