California colloquialisms apply here. When someone is saying "the 101", it probably refers to the Los Angeles segment of US-101. When it's highway 101 or just 101, probably the Bay Area segment.
And the highway names do become useful in radio traffic reports. Even though the names are being removed from signs, and most freeways in SoCal are referred to by their number, thankfully, the core freeways that pass through Downtown LA are referred by name, which helps to differentiate the different sections of road.
I-10: Santa Monica and San Bernardino Freeways
I-5: Golden State and Santa Ana Freeways*
110: CA-110 Arroyo Seco Pkwy and I-110 Harbor Fwy**
* Technically of course, a part of the Santa Ana Freeway is also part of US 101. To my recollection, this is referred to as the 101 freeway in Downtown LA or in East LA and not as the Santa Ana Freeway. Sometimes this section is mistakenly referred to as the Hollywood Freeway. Using the term Hollywood Freeway could be a bit misleading as part of the Hollywood Freeway is US 101 and part of it is CA-170. Similarly parts of the Ventura Freeway are CA-134 and parts are US 101. From my recollection, it seems that Hollywood Freeway is used to refer to US 101 between North Hollywood and Downtown LA, and "the 170" for the part to the north. It also seems that "the 101" is used for the Ventura Fwy section of US 101 and "the 134" for CA-134. So while in the case of US 101, CA 134, and CA 170 the names can be a little confusing, the names work great for I-10, I-5 and "the 110."
** Technically, the freeway between I-10 and US 101 is the CA-110 Harbor Freeway, but generally speaking, aside from that section, I-110 is the Harbor Fwy and CA-110 is the Arroyo Seco Parkway. There is plenty of wrong signage that refers to the section north of I-10 as being I-110.