Are pleonasms like 尷尬 common, where at least two compounds share the same semantic component?
Actually the second character 尬 can be used alone.

和我老婆讨论这种事情真的很尬 = It's really awkward to discuss this thing with my wife

In the example above using 尬 instead of 尴尬 gives basically the same meaning.
The word derives from the pronunciation of 间界 in Chinese Wu, which is &quot;gāngà&quot; (note that tones in Wu are not 100% identical to Mandarin). It appears also in the Mandarin chengyu 半间不界, which dictionaries will list with the pinyin &quot;bàngānbùgà&quot;, meaning &quot;不深刻，肤浅不彻底&quot;.
Then in Wu, the word apparently had shifted in meaning and signifies also abnormality, deviation, etc. and awkwardness.
In Mandarin the characters 尴尬 have been used to represent those sounds (attested as early as the 说文解字) with a better semantical effect, as the semantic component 尢 depicts a person with a limp leg.
In conclusion, even if it appears pleonastic today, the etymology seems to be pointing in a different direction.

As for other words composed of two characters with the same semantic compound and similar in meaning, there's several. A few that come to mind:

坍塌 to collapse (坍 and 塌 both meaning &quot;to collapse&quot;)

忐忑 nervous (忐 and 忑 actually not used alone)

惆怅 melancholy (惆 and 怅 both with the meaning &quot;depressed&quot;)

魑魅 and the chengyu 魑魅魍魉 chīmèiwǎngliǎng, demons and monsters (literally a bunch of 鬼)