The determining factor is who actually paid the tax.

We actually split the cost between the two of us. So does it matter then who claims it?

Then yiu could each deduct the amount that each paid.  Assuming you mean the property tax portion of the registration fee, it is an itemized deduction.  Unless you can otherwise itemize there will be no benefit to claim it.

For 2018 and 2019 many taxpayers that itemized in the past will find that they can no longer itemize because the standard deduction has doubled so all of their itemized deductions no longer exceed the standard deduction.

Only if all itemized deductions exceed the standard deduction will it be of benefit.

Not all itemized deductions count the full amount. Medical expenses are reduced by 7.5% of AGI so if your AGI is $30,000, for example, then only medical expenses more than $2,250 would be an itemized deduction.

The 2018 tax law also caps the total of Sales tax OR State and local income tax, Property (real estate and personal property) taxes at $10,000.

Mortgage insurance premiums. The itemized deduction for mortgage insurance premiums expired on December 31, 2017.

Mortgage interest on loans after Dec 16, 2017 may be limited.

The Mortgage must be secured by the property to qualify.

Interest on home equity loans and lines of credit are deductible only if the borrowed funds are used to buy, build, or substantially improve the taxpayer’s home that secures the loan.

You can check the actual amount of itemized deductions by using the Search Topics for "itemized deductions, choosing" (under "My Account, Tools" in the online versions). Click on "Change my deduction". That will display the actual amount of itemized deductions vs. the standard deduction. (Be sure to uncheck "Change my deduction" after checking it so you do not lock in the wrong deduction.


2019 standard deductions

$12,200 Single
$18,350 Head of Household
$24,400 Married Jointly

Add an additional $1,300 for over age 65 or blind
This amount increases to $1,650 if the taxpayer is also unmarried.