I am not sure that web search is a monopolistic market, though it seems that Google is the dominant player. Google itself wasn't the first player, and managed to capture a large market share from established incumbents. Why do you think that competitors couldn't quickly capture the market? You seem to think this is an obvious fact, but I don't see why, as switching costs are very low.

It seems to me that the important market is online ads (which is where other companies focus), and that Google is the only company willing to invest in web search.

Building a web index and search service to compete with Google requires huge capital expenditures and going up against a network effect that Google exerts on the market (often by paying other companies to be the default). The only company to marginally succeed is Microsoft, and virtually every other search service uses Bing's index on their backend. The cost of switching is low but the barriers to participating meaningfully in the market are high.