Why do cat (猫) and dog (狗) share the same 犭 radical? I have noticed that a cat and dog share the same radical, which is 犭dog (quǎn).

So, if we break the words down:

猫 cat (māo) = 犭dog (quǎn) + 艹 cǎo (grass) + 田 tián (field)

狗 dog (gǒu) = 犭dog (quǎn) + 勹 bāo (wrap) + 口 kǒu (mouth)

I have two questions:

1) Why a dog radical has the dog in it with other components?

2) Why a cat has the dog radical it in?

These words are not difficult to remember. For a cat, I use 'meow', which is somehow similar to māo. For a dog, I stick to 'Go, go, dog!', which sounds a bit like gǒu.

However, I am curious about the origin of these words.
The radical in 猫 actually comes from 豸 zhi4 (beast), which you can still see in the traditional character 貓. 

Whereas the radical in 狗 is in fact 犭 quan3. 

The simplified version of 猫 got the 犭 quan3 radical because of trait reduction and semantic affinity (dog -> beast). 

If it sounds easier to you, you can think of 犭 quan3 in simplified Chinese as generically referring to an animal, as in 狼 (wolf), 狮 (lion), 猴 (monkey), 狐 (fox) and words relating to animals as 猎 (hunt).

As for your other question: 


  1) Why a dog radical has the dog in it with other components?


The dog radical 犭 comes from 犬 which is a pictogram, still used in some words today, especially dog breeds. No radicals other than the image itself. Here you can see an evolution of the picture. (Source: Wenlin, and possibly 说文解字).



If you are talking about the modern character 狗 gou3, it comes from the semantic/radical 犭 quan3 plus the phonetic part 句 ju4. It was probably a colloquial or popular name for "dog" which has made it into the standard language, whereas 犬 keeps a more classical and scientific flavor.