why am showing logged in with like gecko on my yahoo account

Hi. This is really a Yahoo question, not an Apple question, but there's some info here https://au.answers.yahoo.com/question/index?qid=20150421114501AA8ptsB. Apparently, you're using Firefox (Mozilla) on an Apple device, and "like gecko" is a rough estimate of your browser type. They can tell it's Mozilla-based, but cannot identify it precisely.

That is a "User Agent" string - a UA is sent by your browser to let websites know what browser and version you are using (in case they have a browser checker for minimum requirements for their content)

EXAMPLE = Mac Safari

(Macintosh; Intel Mac OS X 10_10_5) AppleWebKit/602.1.50 (KHTML, like Gecko) Version/10.0 Safari/602.1.50

Gecko (software) - Wikipedia

https://en.wikipedia.org/wiki/Gecko_(software)