The short answer is NO.
BizTalk360 supports various versions of BizTalk Server. For the complete list, refer the support article What versions of BizTalk Servers supported by BizTalk360?
With BizTalk360 you can manage multiple BizTalk environments from a single installation as long as all the environment are running on same version of BizTalk server.
Example: All of your BizTalk environments are using BizTalk Server 2013, but you cannot mix BizTalk Server 2010 and BizTalk Server 2013 to same installation of BizTalk360. This is purely due to the limitations imposed by the underlying BizTalk Server Admin API's. They do not allow mixing them, and it makes sense since the underlying database between two versions of BizTalk server may be significantly different.
If you want to manage different versions of BizTalk server, then you need to install BizTalk360 multiple times. Example: One for managing 2009 version and one for managing 2010 version. There are no licensing implications, we allow customers to install it this way. The only draw back is you need to remember two or more URL's.