I need to obtain Rinkeby DAI token with this token address:
I tried a faucet and https://rinkeby.uniswap.exchange/swap - both give me a different token (however with the same token symbol "DAI").
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You can get DAI via compound!
This DAI has this address, not the one you have been looking for, but in 2020 it maybe different:
Just visit compound with metamask being on rinkeby:
I think there's no "DAI faucet", what you need to do is to get ether from a faucet and then you need to interact with DAI contract to send ETH into their contract, and then you'd receive DAI
Here's a guide that might help you: https://github.com/makerdao/developerguides/blob/master/dai/dai-in-smart-contracts/README.md