Yes. This is the difference between phonemic and phonetic analysis.
A particular phoneme such as /ʌ/ might be produced in different ways by different speakers, or in different ways by the same speaker in different contexts. Sometimes a linguist might want to use notation to indicate the difference.
An example of this is "light l" and "dark l" These represent the same phoneme /l/ but phonetically light l is [l] and dark l is [ɫ]. Some dialects and speakers use only dark or only light l. Some speakers use both, but in different contexts. Note the use of square brackets for the phonetic sound, but slashes for the phonemic interpretation of that sound.
The American pronunciation of /ɹʌn/ has a vowel that is rather more forward than the British pronunciation. But this is merely a variation between speakers and isn't significant for the phonemic interpretation of the sound.
So, they don't have the same sound, but this is "just" accent. A learner should normally focus on clear pronunciation in one dialect, and usually either the RP pronunciation of Southern Britain, or the General American accent, widely spoken in America (as these are the accents that most people are familiar with)