It is possible to synchronize your mobile device with Outlook if it is installed on a different computer (without Thunderbird/Sunbird being synchronized on this machine). So you can synchronize your mobile device with Outlook and Thunderbird if they are installed on 2 different machines. Simply be aware that all the modifications performed on either computer will be replicated on the other one. Don't forget to regularly backup your data, especially if it is the first time you synchronize with this configuration.
It will be necessary to configure 2 different partnerships between your mobile device and the 2 machines (where Outlook and Thunderbird/Sunbird are installed). On the machine where you synchronize with Outlook, select "contacts, calendar, tasks or E-mails" in ActiveSync/Windows Mobile Device Center options. On the machine where you synchronize with Thunderbird/Sunbird, select "Cards, Events, Tasks or Mails" in ActiveSync/Windows Mobile Device Center options.
If Outlook and Thunderbird are installed on the same machine and you want to synchronize your mobile device with both of them, then you can get unresolved items during the synchronization. It is generally not recommended to do it, but some users prefer to though. In this case, you may uncheck "Check that a category of items is not simultaneously selected for synchronization with Outlook on the same machine" in "Tools/BirdieSync/Options/Miscellaneous/Checkings".