Do phones with the same camera sensor produce the same image quality?

Note: This is a simplified answer. It does not go into detail, and therefore, please do not take it as authoritative answer on specifics about the exact processes that take place inside a digital camera.

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The same sensor does not necessarily mean equal pictures. The sensor is one component of the whole process - the others are the analog-to-dicital-conversion (ADC) and the digital signalling processor (DSP) with its algorithms.

The sensor is 'just' a converter - it converts incoming light into electricity. The ADC then takes that electric signal and interprets according to a look-up table (i.e. a curve), as sensors are not perfectly linear, so there are dents and spikes in the signal that need to be corrected in order to get a picture that is close to the expectation of the user. The DSP then does de-noising and certain 'image enhancements'.

If any of these components change, the picture might look different.

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Also note that one of the most important factors is the lens: focal length is a very important factor, and image quality can suffer greatly from an (extremely) bad lens.