The compound word ''chairo'' obviously comes from ''cha'' and ''iro'' so you would think that the ''a'' and ''i'' would be pronounced separately. But sometimes it sounds (to me) like they are pronounced as one sound similar to the ''ai'' in the word ''hai''.
Are they pronounced both ways or just one ?
Imagine if you were an English speaker, asked by a Japanese that: "Should I pronounce 'header' as ヘダー or ヘッダー?"
From a native speaker's standpoint, it is actually a non-problem: ちゃいろ is always in three units (morae) cha-i-ro. The fact is not affected by word composition either, as バイト "part-time job" is always ba-i-to too. (And はい ha-i.)
So what leads to your question is, I think, in fact the mismatch between the Japanese tongue and [your native language]'s ear. In every language, fast (I mean, most of the daily) speech compromises on phonological clearness for articulatory smoothness; which means in this case, the hiatus a-i becomes so seamless that many incidental factors such as relative length of a to i, emphasis, or surrounding intonation etc. could make you sometimes hear the diphthong //ai// and other times not. It is hard to make a generalized advice on this because it also depends on your language's threshold when to hear //ai// and //a.i// as syllables.
Either way, no matter whether you believe that "chai-ro" and "cha-iro" are distinct sounds uttered in separate manners, it is just an irrelevant matter of where you accidentally put more muscle for Japanese speakers. The correct way I think is to train yourself to utter cha / i / ro each in same length by default (i.e. when no stress or intonation is in consideration).
It's 茶色{ちゃいろ}: ちゃ-い-ろ. And whether it is pronounced the same as how you pronounce "chai" would depend on your native language. But the first two vowels in the word 「茶色」are definitely not pronounced the same as the vowel in the English words "light", "ride", or "pipe".
The /aɪ/ vowel in English, occurring in the words I listed above, is a diphthong--a gliding vowel formed from long vowels in Middle English in the Great Vowel Shift. It is not a /ɑ/ (how a lot of English speakers pronounce the vowel in "PALM") plus an /ɪ/ (as in KIT). It is a single vowel, not a combination of two equal vowels as is the case in Japanese. (Note: I am an American English speaker, so I am basing most things here on General American (GA), but Receive Pronunciation is also included in this answer, but there are varieties of English where the vowel in bath is rendered the same as the /a/ in /aɪ/.)
Diphthongs are single vowels with some tongue movement that allows a transition in the rendition of the vowel. Modern Japanese doesn't have diphthongs. Japanese is phonetically codified through kana and rendered in distinct syllables that largely follow the consonant-vowel pattern, with each vowel having the same length (which means all syllables are pronounced with roughly similar durations). So the simple answer to your question is: no, it is not pronounced the same way in English.
Count (uniformly) 3 beats: 1(cha)2(i)3(ro). Pronounce it in one continuous flow. (Do not break in between cha and i, or in between i and ro.)
There is not a standalone vowel “ai”. For instance, 愛する takes 4 beats (mora): 1(a)2(i)3(su)4(ru). Pronouncing it in 3 beats as 1(ai)2(su)3(ru) would be incorrect. Likewise はい takes two beats (but in one continuous flow).
