You are not considering the voltage correctly.
You measure voltage between 2 points. So if you got 3 things in series with all the same resistance, A-B-C. Let say we got 6v. It would be 6v A 4v B 2v C 0v in the series.
The voltage between each component is 2v. All the components have the same R so I should be the same through all of them.
The Voltage on A is 6v. The voltage after C is 0v, that a 6v different, 3x bigger, but now you add A+B+C resistance since they are in series, so 3x the voltage over 3x the resistance has you end up at the same current.
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