Algae are a very diverse group and many types of algae grow in freshwater. "Freshwater Algae of North America" by John D. Wehr et al. is a good starting point.

Fresh water algae will grow in fresh water. Salt water algae will grow in salt water. Algae that grow inside of lichens do not need water.

The algae that grow will be the kind of algae that are found near your water bottles. They will need light to grow, because algae photosynthesize with light, to grow.

I googled: algae that grow in drinking water. Here are 3 of the answers:
1. Why is algae growing in bottled drinking water? - Quora.

Apr 1, 2018 - Bottled tap water is more resilient to algae growth because it will contain some residual chlorine. But as with all bottled products, once opened the chances of microbiological growths are increased.

Bottled Water an overview. "There is little or no risk of contamination of water by yeasts, although molds will grow."

The Algae-in-a-Bottle Experiment.

This is an interesting question, and I suggest that you find out yourself. You will need a few small fish tanks, the brands of bottled water you want to test, and a few other things to monitor the growth.

You may need to search the web for places that can provide you with a type of algae - such as spirulina - and the required equipment, then test out the bottled water yourself. The algae will likely take a few weeks to truly grow, during which you will want to keep all the tanks under the exact same conditions except for the brands of the bottled water you're using.

You may also want to measure the mass of the algae by scooping a bit and weighing this bit - just be careful that you're always scooping all the tanks with the same scoop, and that you are getting the same ratio of algae to water so that your comparison of how well the algae grows is valid across the different brands of water. After a few weeks of collecting data on how well the algae grows, you should be able to see the differences, if any.