Good question, Benny. Here’s what I think:

In Received Pronunciation, there is a /ɑː/ phoneme, called the “broad A”. There is another vowel phoneme in words like “Bob” /ɒ/. In General American, these two phonemes are merged into /ɑ/, which has the quality of the RP broad-A, but without the length. British dictionaries mistakenly transcribe the American merged vowel as long, since they are unfamiliar with the version of the sound that is not long. Note: In non-rhotic American accents, the two phonemes are still distinct, and the broad-A is still long. In fact, I believe traditional Boston Accents distinguish the phonemes solely by length.

Phil's explanation is a revelation to me, and it does make a lot of sense.

I think the Cambridge Dictionary is oversimplifying matters by using the same transcription for both the AmE and BrE versions of 'father'. The broad-a in the RP rendition of 'father', usually transcribed as /ɑː/, is not a feature of AmE. The transcription of the AmE version as /'fɑðər/ is much more accurate.

To be honest, Benny, I would ignore Cambridge's transcriptions of American English. They do seem to be filtered through British English ears. For example, if you look at the word 'bob', you'll see that it correctly gives /bɒb/ for the BrE version ( the short, round-lipped and rather "tight" vowel which is typical of BrE and has no equivalent in AmE). But the AmE version of 'bob' is transcribed as /bɑ:b/, because that's what it <em>sounds like to British English speakers. </em>In British English, we do not have the /ɑ/ vowel, so when we hear it and want to transcribe it, we approximate to the nearest vowel sound that we do have, which is the broad-a.

The AmE merger of /ɑ:/ and /ɒ/ into the single phoneme /ɑ/ can occasionally cause problems when we're using proper names. For example, here's an American woman (A) talking to her British friend (B):

A: What was your dad's name?

B: Barney. /bɑ:ni/

A: Bonny? /bɑni/ <em>(Thinking 'That's weird, why did he have a girl's name?)</em>

B: Bonny?!? /bɒni/ <em>(Thinking 'Is she deaf?) </em>I said Barney /bɑ:ni/, not Bonny /bɒni/!

A: Yeah. Bonny /bɑni/ . That's what I said. You said your dad's name was Bonny. <em>(Thinking 'Is she crazy?').</em>

B: No I didn't. I said it was Barney. /bɑ:ni/ B.A.R.N.E.Y.

A: Ohh! I see! Barney /bɑ:rni/

B: Er... yes.

A: Well why didn't you say that?

B: I did.

@Phil Thanks so much for providing the expert opinion.

@Su.Ki. I really appricate how you demonstrate the concept. It's super colorful and useful!

I believe Drasvi’s last comment is entirely correct. By the way, for the record, I usually use // rather than [] simply because the former are a lot easier to type on my keyboard and the difference is generally irrelevant to these discussions. Note that transcriptions usually don’t bother to state all the information explicitly. For example, in the word “pocket” (which supposedly has three "unvoiced stops", more accurately, three fortis stops), the first consonant is aspirated, the second is unaspirated, and the third is (in most accents) usually unreleased and / or pre-glottalized, depending on the accent. None of those details will be found in your typical dictionary, as those phonetic details are systemic variations of the respective phonemes — native speakers will automatically use the “correct” allophone. 

ɒ (upside down) is somewhat rounded.

ɑ is not.

Accordingly /ɑ/ is not guarranteed to be unrounded. It can mean either "those who invented this transcription don't care if it is rounded or not" or "mostly unrounded" or "always unrounded".

When /ɒ/ is used it is likely to be rounded in at least many cases - or why else use this sign?!

I think of these three: /ɑ/ - /ɒ/ - /ɔ/ in most American varieties two or one remain.

Still what is transcribed as /ɑ/ for America may sound as [ɒ] in some accents, and what transcribed as /ɒ/ for Canadian English may sometimes sound as [ɑ] because why a Canadian would care?

(Let Phil correct me)

P.S. /ɑ:/ - /ɒ/ - /ɔ:/ (start - lot - north) are distinguished in British RP accent. I assume when a RP pronunciation is transcribed with /ɑ:/ it is rather systematically unrounded - especially given that RP is one accent among many, while "American English" is a collection of numerous accents. But again let native speakers correct me.