How tightly clustered are the CHIME-detected FRB repeats? In the past few days, journalists have been playing the usual "X could be aliens" game. This time, X is repeating FRBs from a galaxy 1.5 billion light years away. (Or is it 1.5 billion years ago? The proper/conformal time distinction isn't clear in pop sci.) I couldn't find the galaxy's name anywhere, but then I suppose a galaxy that distant might not have been named, even if it's now of some interest. Still, the claim of "repeated" bursts (regardless of their cause) made me wonder: what is our tightest empirical upper bound on how far apart the detected bursts' sources are?

I'm prepared to believe we can see 1.5 billion light years away, clearly enough to pin two events to the same 100,000-light-year wide galaxy, with no others within 2 million light years of it (these are plausible example numbers based on comparing the Milky way to Andromeda). They're small fractions of 1.5 billion light years, but not so small we couldn't resolve an image that clearly. On the other hand, I sincerely doubt we can prove any two bursts that distant are from the same star. I presume what's notable about repeat FRBs is they're so close together in time we expect the source to be common, but I'm curious as to how tightly observations constrain that.
You'd be right to be a little suspicious if CHIME had just detected two events. Fortunately, that's not typically the detection criterion for a repeating FRB. In the case you're talking about, FRB 180814.J0422+73 (Amiri et al. 2019 for the CHIME/FRB Collaboration), six bursts were detected from one particular region, with only 18 total bursts detected in that pre-commissioning phase (so 13 sources total). That high a fraction of bursts coming from a very small part of the sky is enough to raise some eyebrows.
You're also right that CHIME's angular resolution is not enough to localize the bursts to within a galaxy-sized region at this distance ($\sim500$ Mpc), as CHIME folks will readily admit. To lend more credence to a repeating source, long-baseline interferometry is required, involving either follow-up observations or digging through archival datasets. The latter was the mode of choice for FRB 180814.J0422+73, as the group looked at old survey data from the Very Large Array (VLA) Sky Survey (VLASS) and the NRAO VLA Sky Survey
(NVSS). They found five sources in the region with fluxes close to what would be expected from this source. It's insufficient to say which one corresponds to the host galaxy, but it's likely that one of the five is responsible.
At the moment, the CHIME team is trying to improve its localization capabilities by adding &quot;outrigger&quot; telescopes in the United States and Canada, including one at the Green Bank Observatory. Ideally, this would reduce or eliminate the need for follow-up observations with other telescopes.