Is a person considered sinful if one of their family members uses their private perfume and goes out with it, knowing that they did not agree to that, and that this perfume is not designated for use outside the home?

You bear no sin for your sister's use of her perfume without your knowledge, as long as you did not assist her in it. You should continue to advise her with wisdom and gentleness, using appropriate means such as gifting books or seeking help from righteous women, along with frequent supplication. If she does not respond, then speak to her guardian to prevent her from this objectionable act. We also caution that gifting on birthdays and celebrating them are customs of non-Muslims that are forbidden.