Gary Illyes of Google said in the Google SEO office-hours from yesterday that verifying your website in Google Search Console won’t impact your Google Search indexing or ranking whatsoever.
Gary said, “Having your site verified in Search Console or changing the verification code and method has no effect on indexing or ranking whatsoever.”
John Mueller of Google previously said that Search Console verification doesn’t help with crawling either.
Gary added later that Search Console gives you data and analytics that can help you make improvements to your site to help you rank better in Google Search potentially. “You can use the data that Search Console gives you to improve your site and thus potentially do better in Search with your site, but otherwise has no effect on search whatsoever,” he added.
Here is the video embed at the 15:27 mark:

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