The Original Poster has found an enormous typo/display problem in the Cambridge Dictionary. The type of British English described in the Cambridge Dictionary is Southern Standard British English. This variety of English is non-rhotic, which means that we only pronounce R when it occurs directly before a vowel sound. The correct pronunciation is /ʃɔː/, not /ʃɔːr/. If you listen to the audio, you will find that the British English speaker does not have an /r/ at the end of that word.
The Original Poster asks if it is a homophone with the word sure. The answer is that it depends on the speaker. People under fifty are likely to pronounce the two words exactly the same. However, some older speakers use a diphthong in the word sure and pronounce it /ʃʊə/. So for these speakers the two words are entirely distinct.
No. Most (but not all) of England and Wales (but not Scotland) is non-rhotic, which means that a final /r/ is never pronounced unless the word is followed in the same breath-group by a vowel-initial sound.