Your friends’ botany joke is lame. Tree was never taxonomical, only a growth form or “habit” in older parlance. Plenty of individual species grow as trees or shrubs and many genera have representatives from forb through tree. There is no true tree.
The thing is if all you know is that the screwdriver and hammer go in the toolbox, you lack a functional knowledge of those tools and therefore the skill to use them. I for one am confident that you are capable of understanding that while plants and algae are both living organisms, they are functionally, ecologically, evolutionarily, and taxonomically different. The layperson so married to their colloquialisms need not worry about too much detail, yet may benefit from the knowledge that algae are not for bashing nails and plants are not for driving screws. Alas you may wish to know which of the sea you may eat raw or which might seasonally give you a nut flour. Or you may more easily grasp the functional and ecological role of microalgae, armed with the background understanding of their distinction from trees and grasses, should you encounter those creatures. And I hope you do, since you are relying on them to live.
In any case, in my experience, the greater people’s understanding of natural history and their ability to name the organisms in their environment, the more likely they are to see, appreciate, and value the rich mosaic if life around them. The benefits of that are grand for the individual even if they don’t bother connecting the dots on the science.
Colloquialisms are great. But suggesting we all permanently adopt and maintain a state of ignorance because a certain subset of humans had a term, I dunno.
What’s the fish one?