English equivalent to the Bengali Idiom " bamon hoye chand dhorar shokh" which means wish of a lowly person to become equal to aristocratic people

Solution 1:

Consider champagne taste on a beer budget

(idiomatic) Expensive wants or preferences which one lacks the finances to fulfill satisfactorily.

Wiktionary

There is also the saying If wishes were horses, beggars might [would] ride

an English language proverb and nursery rhyme, originating in the 16th century, which is usually used to suggest that it is useless to wish and that better results will be achieved through action.

Wikipedia

Solution 2:

castles in the air

Also, castles in Spain, plans or hopes that have very little chance of happening. AHD says: Extravagant hopes and plans that will never be carried out.
OD offers this sentence example:

  • My father built castles in the air about owning a boat

TFD provides a little history and the following examples of usage.

  • Musing about the bestseller list, she was apt to build castles in the air.

  • She tells me she's planned out her whole career, but as far as I can see it's all just castles in the air.

The first term dates from the late 1500s. The variant, castles in Spain (or chateaux en Espagne), was recorded in the Roman de la Rose in the 13th century and translated into English about 1365.

When people yearn or desire for things that are fanciful, and completely disconnected to their reality, we say the wishes are just pipe dreams

pipe dream

Meaning: An unrealistic hope or fantasy.
Origin: The allusion is to the dreams experienced by smokers of opium pipes. Opiates were widely used by the English literati in the 18th and 19th centuries. […]
The early references to the phrase all originate from in or around Chicago. The earliest I have found is from The Chicago Daily Tribune, December 1890:

  • "It [aerial navigation] has been regarded as a pipe-dream for a good many years."

Source: The Phrase Finder

Solution 3:

"to reach for the moon", "to reach for the stars" or "to reach for the sky", they all mean

. to try to achieve something that is very difficult.

e.g. "If you want success, you have to reach for the moon." TFD

"reach for the moon" (also: reach for the stars) - be overly ambitious; try to do or get something impossible

  • e.g. "She is always reaching for the moon and getting disappointed

You and me, we reached for the sky, the limit was high Never giving in, certain we could win that prize, I should have seen it in your eyes. (You and Me, Frank Sinatra)

Solution 4:

getting ideas above one's station

To get ideas above one's station means to imagine one has the privileges and respect that are reserved for one's social superiors.

station here refers to one's social position. In the glory days of the English class system this was a rather fixed position, hence station.

http://www.macmillandictionary.com/us/dictionary/american/above-your-station