After I typed "a" into Bing, a list of terms that I had recently queried in Google appeared in purple.
How does Bing know what terms I recently Googled?
As @Dez has pointed out, it is due to the Autofill/AutoComplete feature that browsers (not just Chrome; although it may not be activated by default) implement to make filling forms easy.
<name>
isq
just like Google's. Doesn't look like they are taking advantage of local storage, just taking advantage of all the browsers autofill options on form field values of same name.