As everyone has more than one friend, or at least imagines so, it is natural to attach an additional modifier to my friend which informs the listener as to why mentioning this particular friend is relevant to the conversation.
What modifier is useful and appropriate depends on the context of the conversation; this construction has the effect of reducing that person to a single trait. For example, suppose I say
Our water heater is broken, but thankfully my plumber friend is going to look at it.
This makes sense; I have a problem, and a friend of mine with expertise in the area is going to assist me, so I am thankful. Now, that plumber may also be a terrific mother and an accomplished flautist and a devout Lutheran, but those traits are not relevant to the conversation. It would be confusing for me to say
Our water heater is broken, but thankfully my flautist friend is going to look at it.
for what does playing the flute have to do with fixing a water heater? And since knowledge of plumbing is not concentrated in any particular religion, it would be similarly ridiculous to say
Our water heater is broken, but thankfully my Lutheran friend is going to look at it.
Since this construction has the effect of reducing an individual person to a single trait, it has the potential to cause great offense, especially when that trait was historically used to exclude or oppress, or because mentioning the trait recalls demeaning stereotypes.
For example, for centuries there was deeply ingrained institutional discrimination against African-Americans in the United States. Simply being "colored" was enough to exclude people not only from occupations and public services, but indeed the most basic human dignity, and and social discrimination still persists in many parts of American society. To refer to my black friend, then, recalls a time (not long ago) when your race was your destiny, and not the most hopeful destiny for those not of Northern European descent: it denies the friend's individuality in favor of a social construct which should not be relevant, but often still is. And it implies that you sympathize with this past exclusion and cannot see beyond your friend's identity with the group.
For another example, consider the “Polish plumber,” a stereotype which plays on the fear of workers moving from Central Europe to Western Europe to take low-status jobs at lower pay than Western Europeans would accept. To say
Our water heater is broken, but thankfully my Polish friend is going to look at it.
is not merely irrelevant, as in the Lutheran example, but potentially offensive. Saying it in this way recalls the particular association of Poles and plumbing in the trope of the Polish Plumber— as if your friend's nationality implies some innate knowledge in plumbing, and that your friend, again, is nothing beyond her membership in a low-status category of people.
There are situations when one's background is relevant to the conversation, of course, and no offense would be taken or implied.
I'm going to work in Amman for six months, so our Jordanian friend offered to teach me some Arabic.
The Jordanian nationality of the friend is important because the speaker is going to Jordan, and because that nationality implies knowledge of the Arabic language that the speaker would find useful in a tutor.
But since traits like ethnicity, nationality, religion, social class, and so on have the potential to cause offense, a "politically correct" approach is to alter the construction and emphasize the individuality of the friend.
I'm going to work in Amman for six months. My friend Miriam is from Jordan and is a native speaker of Arabic, so she's offered to teach me some useful phrases.