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").
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/dai-in-smart-contracts.md