So how is this different from the current situation with Google dominating the market? Google works in French, German, Dutch, Spanish etc! You only need one company in based one country and they can work in all languages. And if they grow big, they setup offices in different countries, for country specific or language specific searching.
1. Slavic languages have heavy inflection. If Google doesn't understand all the possible forms of a word, it will give you poor results until you search again by changing your query slightly. 2. Google doesn't understand common synonyms, so it doesn't give results that contain synonyms. 3. Russian websites may not care about google rank as much as Yandex, so they might not even be indexed.
I don't know how, but there are thousands of websites I can find by searching Yandex for a certain thing that will not be on Google. It may be those three things combined, or other factors I didn't think of.
French search in French, Germans in German, Danes in Danish, Spaniards in Spanish, Italians in Italian, Dutch in Dutch etc. Then, if your local language doesn't give you good results, you default to English.
Online shopping is mostly a local thing with Amazon as the prominent exception. Even in the market for plane tickets local websites dominate, meaning that a French person travelling from Paris to Berlin is likely to use a different website than a German guy travelling from Berlin to France. Again, the exceptions seem to be mostly American.
European countries even have their own payment solutions.