Does Twine 2 not have an update button or auto-update?
If you are running one of the install-able releases, or the local download of the web-browser base release, then the answer is 'no'. If on the other hand you are using the hosted copy of the web-browser based release then you should be automatically accessing the latest version of that application.
As explained by @HiEv, if you are using one of the install-able releases then simply downloading and running the latest installer will result in you obtaining the latest version.
And same question for formats...
Generally you obtain the latest versions of the Official Twine 2 story formats by updating your current version of the Twine 2 application, which happens automatically if you are using the hosted web-browser based release.
Currently the only Official story format that releases manually install-able interim version updates is SugarCube, these updates can found on those story formats's official web-sites. (SugarCube 2 or SugarCube 1)
The other two Official story formats don't currently have interim version updates that you can download, you would need to manually create/build such updates yourself using the source code found on those projects repositories.
The third-party Twine 2 story formats require manual installation, so you need to handle the updating of them to their latest versions yourself.