Assalam-o-alikum. According to Wikipedia at the time of Umar rad-e-yallah hu, old kiswa was cut into pieces and given to hujaj. Hujaj then used it to shelter from heat like umbrella. Their is no proof that it has any significance in Islam. Indeed it raises doubt in the believer heart, and one should avoid it. You can keep it as an antique but not a blessed item.
Umar rad-e-yallah once heard addressing to the hajer Aswad "if I have not seen Rasoolullah Swallah alay he wa salam kissing you, I would never kissed you as I knew that you are just a stone which can't benefits nor give harm to anyone".
This statement of Umar rad-e-yallah clearly indicated that hajer Aswad can't do anything to anyone, which is part of the wall of Kaaba then how the kiswa piece will do.
May Allah give guidance to each and everyone of us.Ameen.