It seems unecessary to include the school name in the relationship. Just linking by School_ID should be sufficient for matching up records by school.
If you are going to use filtered portals on each tab, you don't need to use the global field, though that is an option. You can set up different filter expressions on each portal on each tab that filter for the sepecific item (floor, ceiling...) without any need to reference the global field.
You might, for example, set up this filter expression for the Floors tab:
Assembly Condition assessment_Type::Assembly_Type = "Floor"
But I am guessing as to what exact value to put in quotes. The same type of filter expression can be used on each tab, you just change the text in quotes in each filter expression.
But this only controls what data is DISPLAYED in the portal. If you are using these portals for data entry, you can enter data for any assembly type on any tab, but if it's the wrong tab, the record will disappear from the portal where you entered the new data and will then appear in the portal of the tabe where the filter permits display of that record.
You may want to put a script trigger on the tab control that updates a global variable with the type specified for whichever panel of the tab control is currently in front. You can then refer to this global variable in an auto-enter calculation to automatically set the correct assembly type of any new record added into the portal on that specific tab panel.
The exact script to use to set the global variable's value depends on the version of FileMaker that you are using.