edited

No it's different, for example when I read a review a music album or an article about music I used to select the text and search directly for something about it on youtube for example, without going in the browser options, simply by selecting the search engine down in the quick search, without changing the default search engine every time.

Thank you

For a while now, Firefox has set all search methods, including the location bar and the page web search you are asking about, to the same search engine as set for the searchbar.

Change the selection used by the searchbar by default (not just for an individual search) and the page search will use that.

• Click on the small arrowhead on the search bar and use the option to change the default search engine to some new choice.
• See Search bar - add, change and manage search engines on Firefox

P.S. This was introduced as a security feature. Adware and malware was changing Firefox settings. This change may be inconvenient for some but gives control back to users. There may be addons that help.

There are countless search related addons.

This one although not specifically allowing you to select google for page search will at least allow different search engines in different methods.

• Keyword Search 1.0.8.2 https://addons.mozilla.org/firefox/addon/keyword-search

• And here are the countless (Well a few hundred depending on parmeters used) addons https://addons.mozilla.org/en-US/firefox/search/?atype=1&cat=13

In this case , this addon may be useful...

https://addons.mozilla.org/en-US/firefox/addon/instantfox/?src=cb-dl-popular