Preferences stay with the browser
Saved titles, manual episode checkpoints, recently opened titles, appearance and viewing-country choices use browser storage. They do not require an email address. Different browser profiles can therefore show different lists. If you use a shared profile, other people using it may see the same local choices.
Movie images and viewing destinations
Catalog images are requested from TMDB image servers. Following Watch now opens JustWatch, which has its own privacy practices and may lead to another viewing service. Those requests are not made anonymous by the fact that the initial movie search required no account.
Use the relevant controls
Manage saved titles, checkpoints and recent history with the controls provided for those features. Preserve a list backup before intentionally clearing browser data if the list matters to you. Do not put passwords, payment information or private account details into the movie search or list-import field; neither feature needs them.