If you work in multiple states and pay taxes in multiple states should the sum of your individual state taxable incomes agree with the amount of your federal taxable income?
I believe so.
No - even if you work in only one state your State taxable (or gross) income will virtually never equal your Federal taxable (or gross)income. For many reasons. Like State tax is deductible from Federal, but not from itself! Some contributions or types of income too, (like retirement plans, some types of SUI, FICA, etc.), are deductible for one of the taxes, not the other. Etc.
In the multi state sceanario you mention - also consider - if you had income in 3 States...one of which doesn't even have an income tax. Hence, no State Taxable income. The remaining two incomes better not equal your Federal.