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 "gāngà" (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 "bàngānbùgà", meaning "不深刻，肤浅不彻底".
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 "to collapse")
忐忑 nervous (忐 and 忑 actually not used alone)
惆怅 melancholy (惆 and 怅 both with the meaning "depressed")
魑魅 and the chengyu 魑魅魍魉 chīmèiwǎngliǎng, demons and monsters (literally a bunch of 鬼)
This kind of words is called 聯綿詞 (words that cannot be interpreted character by character). Its antonym is 離合詞. There is a survey in Chinese talking their origin. Interestingly, it seems that they gradually share common initials, final consonants or common radical. 聯綿詞 can be categorized into the following types:
- Both initials are the same: 琵琶, 坎坷, 尷尬, 唐突 etc. (see more here)
- Both final consonants are the same: 荒唐, 須臾, 逍遙, 蹉跎, 腼腆 etc. (see more here)
- (1.) and (2.): 孑孓, 輾轉 etc. (see more here)
- Don't fit into the three above: 鸚鵡, 嘀咕, 蜈蚣, 囹圄 etc.
- Doubtful, but in the same spirit, loanwords: 咖啡, 馬達, 吉他 etc.
- Reduplication: 翩翩, 姍姍, 落落, 泱泱 etc (see more here)
Note that some words in category 4 are in fact in 1, 2 or 3 when you read them in ancient Chinese.
It worth noting that there is even a dictionary entitled 聯綿字典, edited by 符定一 in 1946, which contains more than 23.000 entries of 聯綿詞.
It is not unusual that the 2 characters used in a term have same or similar meaning. This, in my opinion, is because the majority literature works during the peak period of Chinese literature, i.e., 唐 宋 元 are all strictly formatted. When a writer wanted to present a concept, but unfortunately, the format required 2 characters (good example: 七言 律诗/绝句 usually is 2-2-3 format), he had no choice but use 2 characters with same meaning. The amazing part is sometimes they are all under the same radical, top No. 1 is radical of 鬼，dozen of it with similar meaning, and double 王 on the top: 琴瑟琵琶。 Comparatively, the most popular format in English, the iambic pentameter, is really very very loose.