Your words: "Every memory your brain has ever produced is still there [..]"

How would that not imply infinite storage?

It wouldn't imply infinite storage because human life is not infinite in time and memories do not accumulate at an infinite rate in storage consumed per unit time, so the total storage over a human lifespan is finite, so the claim can be true with finite storage.

It is almost certainly false, but it doesn't require infinite storage to be true.

> human life is not infinite in time and memories do not accumulate at an infinite rate in storage consumed per unit time

Which would put it into the category of the second part of my comment--which the person I was responding to said was not relevant to what they meant.

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