Verifying is separate from deploying. One should not affect the other.
If the same proxy contract or related proxy contract was already verified on the network, the hardhat-upgrades verify task may show that the contract is already verified and Etherscan would show "Similar Match Source Code". You can consider this as a verified proxy.
If you need your proxy to show as an exact match rather than similar match on mainnet, contact Etherscan as described here.