No. Subdex is an independent tool with no relationship to Reddit, and Reddit trademarks belong to their owners.
What that means practically
The data is not from Reddit. It comes from independent archives that collected public Reddit content. Records may differ from what Reddit shows today.
No Reddit account is involved. Subdex never signs in to Reddit and never asks you to.
No private content is accessible. Private messages, private subreddits and anything requiring authentication are not in these archives and are not reachable.
Reddit's terms still apply to the content. Posts and comments belong to the people who wrote them.
Opening the original
Where a record has a permalink, Open on Reddit takes you to it in a new tab. That is a link you click, not a request Subdex makes — it does not scrape Reddit to check whether content still exists.