Are the bacteria doing the same photosynthesis as plants do?
Photosynthesis is always about using solar energy to oxidize a mineral molecule and reduce an organic molecule, which becomes an energy source for the photosynthetic cell.

Bacteria and plants have different types of photoreceptors. It is not always the same chlorophyll, maybe even no chlorophyll at all ! The different colours in algae come from the different pigments they are using to do photosynthesis.

Plant cells carry out photosynthesis in chloroplasts, which are organites, whereas bacteria carry it out in their plasmic membranes.

Plants do aerobic photosynthesis, which means they reduce dioxygen into water. Some bacteria do anoxic photosynthesis, which means they can reduce hydrogen sulfide into sulfur.