There are several options to achieve this with SpiraTest / SpiraTeam:
1. You can simply create a set of custom properties to store the different platform components (OS, database, browser)
and just assign these as Test Run custom properties.
Then when you execute the tests, you can specify the appropriate value(s).
2. To be able to more easily report on the results, you can extend option (1) to also add the same custom properties onto the Test Set artifact type
Then you can specify the platform custom properties on different combinations of test set and assign the appropriate test cases to each test set. During execution, the test set will auto-populate the corresponding values on the test run(s).
3. An alternative if you need to track the test cases separately against requirements (i.e. Windows and Linux must both pass to have the requirement fully tested) you can simply create a generic test case that contains the test steps and then just create additional test cases for each platform combination and then use the Insert Link feature to just add a link to the generic test case as the sole test step of each of the platform-specific test cases.