Your question is a bit unclear. If you mean that the last two letters in 'has' sound the same as 'as', you are right. I first thought that you meant that the letter 'h' is not pronounced at all. But that is only the case in French and Spanish. There the 'h' is silent. The pronunciation of the 'a' in American and British English differs a little though. But that for both words I think.

Great question, Hoon. Most native speakers are unaware that in normal speech, the "h" is usually only pronounced at (1) the beginning of a phrase or (2) at the beginning of an accented syllable. The word "has," as an auxiliary verb, is unaccented, so the "h" is dropped in the middle of a phrase. Sometimes, this is written as 's but even if the full form is intended, the "h" will usually be dropped in pronunciation. This is true in US, UK, and other standard varieties of English. This has been true throughout the entire history of English, as witnessed by the spelling of words in Middle English, as well as a comparison to other Germanic languages. (For example, the English word "it" is the Dutch word "het.")

This is not an issue of British or American pronunciation. As Phil says, It's a question of whether the syllable is accented or not. Whether they realise it or not, all native speakers, wherever they come from, automatically 'drop' the 'h' in words such as 'has' in natural speech in the middle of a phrase. The vowel sound also changes, and becomes a 'schwa' sound. This is why 'You should have come' actually sounds like 'You should of come' when spoken naturally by a native speaker.

In British vs American English the H at the beginning of words is treated differently. I'm not certain if there is a pronunciation difference with the word "has" but in the US "has" has an H sound at the beginning. It sounds similar to "as" but with the H sound.

By the way, the dictionary entry you posted clearly indicates three pronunciations of "has," one of them without the "h" sound: /həz/ /hæz/ /əz/