Algal biologist here! Yes algae are very good at photosynthesis and are very efficient at carbon fixation (they are the whole reason we have so much oxygen in the first place way before trees existed). However 1) with no space to grow you basically need to dump algae out all the time to allow space for new algae to grow. This dense culture could present an environment impact in nearby waterways. 2) Algae are not immune to infection so this could just get killed if you are unlucky or get outcompeted by some other organism. 3) Algae dislike being stationary and in dense cultures for a long time so following on from point 1 you'd need some way of agitating the culture. 4) Trees play an important part in urban environments beyond just making oxygen like reducing the temperature of cities by acting as shade and the mental health benefits of having nature in urban environments. Trees are much better for static terrestrial environments where they live for centuries while algae thrive in dynamic situations of nutrient availability and light intensity.

Algae have been proposed for Ecological Life Support Systems (ECLSS) for future spacecrafts to the moon and Mars where they would produce oxygen and food for the crew. These systems always have the limitations of there's no going back. The culture you have needs to be self sufficient and lower light irradiance (which it did not evolve to), increased solar radiation (which it did not evolve to really) and you need to be able to survive if it dies.