More specifically, is there a historical reason why some katakana characters look similar to the hiragana ones, as the question suggests?
It is just a coincidence. As you (probably) know, both hiragana and katakana came into existence as shorthand for kanji. Here's the graph shown on Wikipedia. So you can see that ラ and う are derived from different kanji and just so happen to look similar.
Actually, I didn't know it was shorthand. That's awesome! May 24, 2017 at 21:33
To expand on this note, Japanese had no written form until the Chinese writing system was adopted via Korea, and writing was done by translating the Japanese into classical Chinese; this was until the use of Man'yougana, which was a syllabary consisting of kanji used to represent associated sounds (often more than one kanji per sound); this led to hiragana, whose shapes were based on the overall shape of a particular kanji, and katakana, whose shapes were based on the shape of a component of the same kanji. May 24, 2017 at 22:05
There is also the issue of Joudai Tokushu Kanazukai, distinctions between two classes of manyougana that distinguished a number of sounds, but the distinction was lost when manyougana stopped being used in favor of kana, which did not differentiate between the two classes of sound. May 24, 2017 at 22:09