handkerchief, kerchief, shawl
帨 centers on a piece of cloth used for personal care or covering: a handkerchief, kerchief, or shawl. The common thread is a soft fabric item carried or worn close to the body.
帨 combines 巾 (cloth) with 兑, which likely contributes a phonetic role. The character is associated with fabric items like handkerchiefs and shawls, but the exact historical development of the right-side component is uncertain.
The left side 巾 is a cloth, and the right side 兑 looks like a person holding something. Imagine someone using a soft cloth to dab their face: that cloth is a handkerchief or shawl.
For セイ, picture a person saying 'say' as they offer you a handkerchief: 'say' -> セイ, and the cloth is handed over with a polite word.