If you are using the Java version of Selenium2Library, there is a keyword you can use.
${caps}= Get Remote Capabilities
${version}= Set Variable ${caps['version']}
To do this with the Python Selenium2Library, you have to call a private method since there is no Get Remote Capabilities keyword:
${s2l}= Get Library Instance Selenium2Library
${caps}= Set Variable ${s2l._current_browser().capabilities}
${version}= Set Variable ${caps['version']}
The browser name is stored in an entry in the capabilities dict with this key: browserName
Use Collections.Log Dictionary to have a nice view of the capabilities dict in your log.