Sure, the people who created a religion around him weren’t quite sub-Saharan Africans (although they likely had distant ancestral connections from there), but why should that matter unless the notion is that a deity matches the complexion of the people who believe in it? Which would in effect suggest that people create deities, not the other way round - and that is something no devoutly religious person could ever admit to.
Since neither of us are religious, Yahweh exists only in the minds of his worshipers, therefore he is defined by their cultural biases. The Hebrews probably conceived him as being vaguely Hebrew-looking. Are my jokes so terrible that I need to start marking them with smiley faces? 
If Jesus was terribly exotic looking to his contemporary audience it probably would have been mentioned.
To suggest that the Persians, Arabs, and everyone else in the region are just tanned Caucasians is completely absurd and reflects extreme ignorance and a distinctly Eurocentric racist mentality.
They are still Caucasian because as you so correctly stated white is not a race. As a matter of fact, the Caucasus is just north of Turkey and Iran. It is the people there who are the Caucasian archetype (hence the name) not blond, blue eyed northern European types. To suggest otherwise is not only Eurocentric racism but etymologically absurd. East Asians get darker as you approach the equator too, it doesn’t take long to select for a few pigmentation genes when the alternative is perpetual sunburn.
