This isn't new, it's a longstanding bug / side effect of the feature (developed under the code name "1993") that when you start typing a search it jumps you up to the omnibox. Basically when you click on the search box in the NTP it actually focuses the omnibox but visually shows the search box below as focused.

I'd love to see this fixed or possibly even the whole approach revisited, but it's not clear who owns this now.

Granted I rarely open a new tab page then tab to the search box in the middle of the page (since focus defaults to the omnibox), nevertheless I find the behavior extremely surprising.

The NTP is now owned by the Zine team. I agree that this behavior isn't ideal. But for now this is WAI. 
I'll leave this report open so that we can revisit it once we're going to touch this part of the NTP.

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Marc, will this change in the local NTP? If so, this is basically obsolete. If not, we can still consider archiving it as it's not a priority.

This stays the same on the local NTP. I guess the original reasoning was that the omnibox is typically focused by default anyway, so there's little reason to tab to the fakebox. I do agree that it's a bit weird/surprising.

Let's keep it around as one of the things to consider in the next NTP redesign.

Cross-ref: There's at least two other bugs for this,  bug 696239  and bug 778489. The latter in particular has more information.

Issue 447822 has been merged into this issue.