Learn English – Origin and pronunciation of “Thawte”

The /ˈɛryən/ pronunciation is just a result of English phonology processing a foreign borrowing that starts with the letters AR.

Aryan is a borrowed word in all languages outside the Indo-Iranian subfamily of Indo-European. The rest of the world pronounces it as some variant of [arjan], which comes, as noted, from Sanskrit ārya /a:ryə/ 'compatriot'. Therefore, /'aryən/ is a perfectly acceptable English pronunciation, and the only acceptable one when using the term in its modern Indian sense.

Any use of Aryan (outside scare quotes) that refers to Germany or white racism is a result of romantic interpretations of 19th century German linguistic scholarship (e.g, Grimm's Law), which unearthed the prehistory of the "Indo-Germanic" (as I-E was then called, from names of its Eastern- and Westernmost families) languages. It was all very exciting, apparently. See also Wagner, Mad King Ludwig, Neuschwanstein, German Empire.

The AHD of IER says that Skt ārya comes from the PIE root *aryo- 'Self-designation of the Indo-Iranians'; other descendants of the same root are Iran and, surprisingly, Eire -- Celtic languages sometimes retain PIE roots that are otherwise lost in the Centum group.

None of these are English words, and so English treats them the same way it treats all borrowed words -- it changes the pronunciation until it tastes right. That's all.

Edit:

I almost forgot, another reason to pronounce Aryan /'aryən/ is because Arian /'ɛriən/ usually refers to Arianism, a very important variety of Christianity that was the religion of the Ostrogoths in Italy, and the Visigoths in Spain. The only Gothic texts known to exist are translations of various parts of the (Arian Christian) New Testament.

There is a universal (more or less) system for recording human speech sounds.
It's called the International Phonetic Alphabet, or IPA.   (Note: That's Phonetic, not Phonemic.)
This contains all the sounds that occur in any human language.
Naturally, no human language has all of them.
No human language even has most of them.

Individual languages use only some of those sounds, and representations of words in individual languages use a special selection of these symbols -- different for every language -- called "phonemes". Usually the symbols used are taken from the IPA, but they represent only the sounds and their distribution and pronunciation in that language.

For instance, this is the American English phonemic system, from Kenyon and Knott.

English dictionaries published in the United States normally don't use phonemic pronunciation, preferring the system invented by Noah Webster, based on spelling instead of phonetics. Webster was a spelling reformer, and believed that the traditional English spelling could be used to indicate phonetics. He was wrong, but that hasn't influenced American dictionaries.

English dictionaries published elsewhere, or bilingual dictionaries, or dictionaries intended for language learners, normally do use standard phonemic transcription. If you have a bilingual dictionary, look at the pronunciations in the English part; they will normally use either the system of Kenyon and Knott (American) or a system of RP (UK), which has some differences from American. This is because English speakers normally pay no attention to the pronunciations in the English part, but English learners do, and they need accuracy.

The example cited

\pȯrt-ˈman-(ˌ)tō\

is the style used by Merriam-Webster, based on the original nonphonemic transcription.
In Kenyon and Knott's system, it would be

/portmænto/

and in RP it would be

/poətmæntəʊ/

Regardless of what the Wikipedia entry above says, the M-W system is not phonemic.