Flask's default session stores the whole payload in a cookie, serialised as JSON and signed with SECRET_KEY using itsdangerous. Signing prevents tampering; it does not prevent reading. Anyone with the cookie can base64-decode the first segment and see every key you put in the session.
Confirm it in one line by copying the cookie value and decoding the part before the first dot. Nothing about the value looks obviously readable, which is why user identifiers, roles, and occasionally tokens end up there.
Store an opaque identifier and keep the real data server side, using a server-side session extension backed by Redis or the database. Independently, set SESSION_COOKIE_HTTPONLY (on by default), SESSION_COOKIE_SECURE = True and SESSION_COOKIE_SAMESITE = "Lax", and rotate SECRET_KEY with SECRET_KEY_FALLBACKS so existing sessions can still be verified during a rollout. Note also that browsers cap a cookie at roughly 4 KB, so a large session silently fails to round-trip rather than raising.