What made twitter differentiated was that its tags can be arbitrarily created by the users of the APP. This seems innovatively different from other websites at the time where the tags are pre created and fixed.
According to this article, Twitter didn't invent the hashtag. Of course this doesn't mean the article is authoritative, but this article and this Wikipedia entry suggest that the hashtag was first proposed by Chris Messina publicly in 2007. Since it was proposed publicly, it would not then be patentable. From Wikipedia:
Messina made no attempt to patent the use because he felt that "they were born of the internet, and owned by no one".
Twitter seems to have no hang ups about pursuing patents as I can find 1,483 granted patents assigned to them as of 9/7/23.