yeah i also face the same problem, cookies not store in local storage, i work with nuxt and nestjs. solve by send domain of frontend in cookies

 .cookie('token', token,{
    domain:"web.frontend.cloud",
    sameSite: 'none',
    secure:true,
    httpOnly: false,
  })

after add domain:"web.fronend.cloud", the cookies finally stored in localstorage/cookies

for more info how domain accepted read here https://datatracker.ietf.org/doc/html/rfc6265#section-5.1.3