I really didn't want to get into this whole Clippers uniform argument, but I do have to point out that being in the same building currently doesn't necessarily mean that they cannot represent different areas of the city.
For a little history on the subject which you may or may not already know, the Clippers moved from SD to LA in 1984 and played their games at the Los Angeles Memorial Sports Arena in Expansion Park. Expansion Park just so happens to be 10 miles from Compton and less than 2 miles from South Central LA. One could argue that this proximity to these areas drew a different fan base to the Clippers than the Lakers, who already had an established fan base and were known to be very "Hollywood" with the Showtime era.
The Jets and Giants share a stadium and the Jets have always been known to represent Brooklyn, Queens, and Long Island, whereas the Giants are usually associated with Manhattan, the Bronx and Staten Island. This clearly stems from the fact that the Jets played in Shea stadium for 20 years and the Giants did the same at Yankee Stadium. But it's that history that plays a large part into why their fan bases are what they are today, even though they currently share a stadium.
In Italian Soccer, AC Milan and Inter Milan share a stadium and they have a VERY different fan base. The same could be said for Roma and Lazio.
I get your point, and I would have to agree with you that the Clippers don't necessarily scream "Compton" at all, or even more so than the Lakers. I just don't think citing the shared arena is a good reason why.