I think it's possible, but this is extremely difficult to measure. Measles isn't a novel virus, and immunity against measles seems to be quite robust (thankfully, since it is not uncommon for measles to destroy your immune system and make you vulnerable to everything which you previously were immune to!).
For coronavirus, it's difficult to determine whether increases in rate of spread and how quickly a variant takes over from other variants are due to shorter serial interval, or greater transmissibility, or immune escape characteristics of the new variant.
Remember that the original virus spread extremely quickly as well (Wuhan & NYC come to mind). We had similar levels of mitigations in place over Christmas & New Year's this past year, so you'd expect rapid spread in the absence of immunity (which was definitely in short supply!). And indeed the results seemed similar. (I'd guess a similar (same order of magnitude) number of people in NY were infected in the initial wave - perhaps as much as half the number as were infected by Omicron? I haven't really gone back and dug into the numbers, but under-ascertainment was huge in that first wave in NYC. WAGs here.)
I think it's fairly clear that Omicron is considerably more contagious than the original given the substantially higher numbers of cases observed in most places...but as contagious as measles? Tough to make that comparison at this time. Maybe in a few months or a year, after there's been some time to study spread in various different scenarios, we'll have a better idea of the exact numbers. I'd guess an R0 of 6 or 7, but really a total WAG.
Fauci is optimistic at least, though note that he doesn't like to talk about "endemicity" - seems to me we have a ways to go before we can talk about that and what that might look like. Malaria and tuberculosis are also endemic. Hopefully we don't have that sort of situation!