The short answer: You can't associate Techsmith Snagit with your Apple ID.
The complete answer:
There are two types of software (from the point of view of Apple):
Firstly, those purchased from the Apple App Store. These are associated with your Apple ID and only require your Apple ID (and password) to be installed on other Macs. An Apple App Store licence allows you to install the application on multiple Macs (up to 5) associated with your Apple ID.
Secondly, those not purchased from the Apple App Store. These can not be associated with an Apple ID. Installation on multiple Macs is usually a manual process unless you have some enterprise grade distribution software. Licensing conditions will vary and you may or may not be permitted to install on multiple Macs. Also it varies by product as to whether and how the license is enforced on multiple Macs.
Thirdly (did I say there were two types?), there are special cases which Apple can use for its own software. The iWork 09 suite is a notable example. The Apple Store software does, as you have discovered, detect this separately licensed software and offer you an upgrade (for free) to the latest visions of Pages, etc. It also offers to upgrade trial versions of iWork 09 (but that is another story).
One of Apple's purposes (apart from making money) in creating the Apple Store was to make it simpler for users to keep track of their purchases - exactly the situation in which you find yourself. I track mine by 'printing' all my license and purchase emails to a single folder in my Documents folder - at least they are all in one place. You could improve on that by keeping them in a Dropbox folder so that they are available on other computers and in the event of some disaster to your Mac.