Yes, miners get a reward for "re-confirming" blocks; no, it's not a direct reward.
Each block must reference the preceding block by including a hash of the preceding block's header in its own header. That means each block confirms the block before it, which confirms the block before it, etc..., all the ways back to block 0 (the genesis block).
When we say a transaction has 1 confirmation, that means it's included in a block.
When we say a transaction has 2 confirmations, that means it's included in a block which is followed by one block.
When we say a transaction has 3 confirmations, that means it's included in a block which is followed by two blocks. Etc...
When the block subsidy stops (or becomes so low it's mostly irrelevant), miners will create blocks only for their transaction fees. So miners will attempt to mine block A for its fees (giving its transactions one confirmation); after it's mined, miners will attempt to mine block B for its transaction fees---and block B will confirm the transactions in block A, raising their confirmation score to two. So transactions will still get more than one transaction then.
There is some concern that Bitcoin with transaction fees only will encourage miners to fork the chain and "re-mine" recent blocks with high transaction fees. Some people call this "fee sniping", and if it's a legitimate concern, it still probably won't manifest for several years---and developers are already talking about ways to make it harder and less potentially profitable.