When the last BTC is mined around 2140, what happens to the whole ecosystem? I mean, there won't be anymore incentives for the miners to verify and confirm transactions, right? The only reason miners confirm the transactions in the whole BTC ecosystem is to be rewarded with more BTC. No more BTC reward, no confirmations. Thoughts?
The incentive will come in form of transaction fees.
Right now the block reward is the lion share of mining profits. Within 3-4 halvings however, you can expect transaction fees becoming the main mining income. This also leads to contentious hardforks such as we see today to become harder and harder to pull off, because in the end actual transaction throughput will define which chain is the most profitable to mine.
You can search PoS(Proof of Stake) for answer that you're looking for.
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Proof of stake is a wholly different approach and has nothing to do with BTC.