Would you say the same thing applies to message boards?
It depends on the kind of information that is gathered about you and how this information is used. I suggest always reading the privacy policy as it's all in there. To use this forum's policy as an example, it lists what is collected about you: your username, e-mail address, IP address, and whatever else you choose to share (i.e. in your bio or posts). Obviously your posts will be public but your e-mail and IP address are only accessible to the site admin. However, this is what we would call "personally identifiable information" as, depending on whether you use your real e-mail and IP address, or even profile picture, it can potentially be traced to your real identity.
Other websites might go so far as to download extra information such as your entire browsing history, open tabs, and even bookmarks. The information collected about you can also be sold to third parties for "marketing purposes" as is often stated in the privacy policies of many mainstream platforms. Or, it can be collected by third parties themselves via advertisements and cookies which allows for "cross-site tracking". To give you another example, a random website you visit might have a Facebook tracking cookie because it contains a "Share on Facebook" button; that cookie will collect information about you such as your IP address, now if you have a Facebook account connected to your real identity and you use the same IP address to log into said account, the FBI (Facebook, Inc.) will know who was browsing that website.