Flask – multiple session cookies with the same name
Without code, it is hard to tell where Flask generates multiple cookies with the same name, but it is possible that you let the cookie live to long, please see Flask: How to remove cookies? to deal with this issue.
If I understand you correctly, another main issue is [comment by me]:
Flask picks a wrong session cookie [of multiple ones with the same name]
Picking the wrong cookie from multiple ones with the same name is not Flask-specific, but rather about the logic used by client/ browser, see e.g. How to handle multiple cookies with the same name?