Does my tuition amount for 2019 only apply to the calendar year (ie. January 2019 to December 2019) or the college year (September 2018 to May 2019)?

Calendar year (January 2019 to December 2019).

The amount in box 1 of the 1098-T is the amount received by the college, in 2019.  For the education credit, or deduction, or for a 529 plan distribution, you may only count money paid in 2019.  Furthermore, the money paid must be for 2019 classes. There is one exception, you may count money paid in 2019 for the first term in 2020. 

For further clarification:

 - Qualified education expeneses are claimed in the tax year they are paid. It *does* *not* *matter* what tax year is paid *for*.

 - Scholarships, grants, 529 distributions are claimed as taxable income (initially) in the tax year they are recieved. It *does* *not* *matter* what tax year that money is *for*.  The taxability of this income will be offset by the "qualified" education expenses they are used to pay for, if paid in the same tax year you received it.