I'll explain, because it is true that I'm using "producer surplus" in a non-standard way, but I'm doing it to illustrate a specific point about how Google's business model works and why it is so profitable.
Normally, in an Econ 101 supply/demand model, you have an upward-sloping supply curve indicating how many units of some commodity a producer is willing to sell at a given hypothetical price, and you have a downward-sloping demand curve indicating how many units of some commodity a consumer is willing to buy at a given hypothetical price, and the point at which they meet is called the "market clearing price" where supply and demand are in balance. The area above the supply curve and below the market-clearing price is called the "producer surplus" [1], the amount of extra money the producer gets from the gains of trade. Very roughly, you can think of it as operating profit, though it's not quite the same thing because operating profit is a real tangible dollar amount while the supply curve is largely a hypothetical.
The way Google works is that it comes in for each individual transaction and effectively tells the advertiser "So you have a product or service that you'd like to sell? How much is it worth to you?" It runs a second-price VCG auction [2][3] to determine which of the available ads are shown to the user. Why second-price? Because an ordinary first-price auction incentivizes users to bid less than their true willingness to pay, because they know that if they win the bid, they will have overpaid. Same issue as buyer's remorse in real estate transactions: you know that if you won the bid, it was because nobody else thought the house was worth as much as you did. With a VCG auction you're only paying the marginal harm to the next bidder of winning the auction, so you have an incentive to bid your true valuation. And this is why I used the term "producer surplus" in the original post: it's to capture how Google effectively elicits from each advertiser the maximum amount that the transaction is worth to them, which if they're bidding rationally is the difference between their reservation price and the price they'll receive from the transaction, i.e. the producer surplus.
This also demonstrates where I've been playing fast and loose with the terminology. The price a VCG auction participant pays is not actually their bid, it's the marginal harm caused to other participants (basically the sum of all bids except the advertiser, minus the sum of bids of the other winners). But in a fairly competitive market, one with multiple producers who all face roughly the same cost structure, you'd expect that this quantity converges to the advertiser's actual bid, leading to a condition where Google has actually captured the entire producer surplus and maximized its revenue.
[1] https://www.intelligenteconomist.com/producer-surplus/
[2] https://en.wikipedia.org/wiki/Vickrey%E2%80%93Clarke%E2%80%9...
[3] Technically, it's a modified VCG auction, because the actual ranking score is a machine-learned function that also includes factors like "What's the likelihood the user is going to click on this ad?", "Is this ad spam or malware or illegal?", and "Is there brand damage to us from highlighting this ad?" But from an economics perspective, the only important part is that it's a second-price auction.