I don't think it has anything to do with shame. If you were to ask most of these people if they were Italian, they'd say they were, and express pride in it. They just don't 'claim' it per se in their day to day lives.

There are no Italian communities in California. Some older Italians think there are, because they meet up with a few other families at whatever last standing mom and pops Italian eatery is in the area, but these are really more like historical clubs than any real pillars of preservation.

A lot of Mexican-American neighborhoods in California are former Italian neighborhoods. Interesting how it worked out that way. Many of the remaining Italians that grew up in these neighborhoods identify as Chicanos.

I also wouldn't confuse Chicano culture with Mexican culture. Chicano culture is mostly English speaking, and much of the music Chicanos like is the same that older Italians back east like, oldies, r&b ballads, etc.

Catholicism is also certainly a major commonality between these two cultures.