What part-of-speech would a vehicle's year/make/model be?

"2003 Ford F-150" is a string of adjectives describing a truck. Trademarks are always adjectives, describing a noun by specifying its manufacturer or originator.

As the International Trademark Association puts it:

Trademarks and service marks are proper adjectives. Not nouns. Not verbs. A mark should always be used as an adjective qualifying a generic noun that defines the product or service. A mark is a company brand name, not a product or service itself. -- A Guide to Proper Trademark Use

It's much clearer if you say "2003 Ford F-150 truck" that "truck" is the noun and the other words describe what kind of truck it is. But it's perfectly grammatical to omit the noun when it's clear from context what the noun is.

Ford F-150 is the official full name of the car (make + model). Grammatically speaking, it is a compound proper noun:

I own a Ford F-150.

F-150 is an abbreviated version of that name:

I own an F-150.

So is Ford in the following statement (although the designation could refer to any Ford motor car, not just an F-150):

I own a Ford.

In your example sentence

"I own a 2003 Ford F-150",

2003 is an adjective; it is not a necessary element for specifying the type of car, but describes one of its attributes (namely its age).