Huh! The Brooking's story actually cites this Gallup data. But it's a little out of date. Looks like Republican ID has sprung back up in the last ~6 months. Touché!

It's really noisy poll-to-poll; really, I'd be skeptical of anyone claiming that there is much of a meaningful trend based on it. Pew’s series which separately tracks partisans and leaners might be more useful for spotting trends, but they do take seem to have really recent data public in that series.