With me being in the military and my partner who is not my dependent, relocated with me to my permanent duty station. Does his flight and car shipping also deductible?

No.  Your partner's travel and vehicle transportation costs are not tax deductible.

His moving expenses are not deductible on his federal return. 

With the change to the moving expenses tax deduction, only seven states allow moving deductions or exclude moving expenses from taxable income:

• Pennsylvania
• New York
• Massachusetts
• New Jersey
• Arkansas
• California
• Hawaii