Why do we require a CSPRNG's output to be indistinguishable from true random?
We simply strive for crypto that's as close as possible to ideal. Indistinguishably is the strongest property we can demand from a PRNG/streamcipher.

It's hard to predict which non ideal properties will lead to problems at some point in the future. For example the non ideal properties that lead to padding oracles, BEAST, CRIME or the RC4 biases were known for years before they were shown to be exploitable.

Many security proofs assume indistinguishability, so if the PRNG is distinguishable, this weakens the proof. If the PRNG is distinguishable under certain conditions, even if it's a seemingly harmless distinguisher, then a proof relying on indistinguishability becomes useless if those conditions are met.

Consider a message encrypted using a block-cipher in CTR mode. This has the property that no blocks in the key stream will be identical (just like Fortuna/Yarrow without re-keying). When the ciphertext contains identical blocks, the attacker knows that the corresponding plaintext blocks were different, because the corresponding key-stream blocks were different as well.
Admittedly that's not a huge leak in most contexts, but it is a leak. It might lead to problems in some contexts, for example if the message only uses a few different values for each block.