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In Captain America: The First Avenger, during the final one-on-one fight between Captain America and Red Skull, Red Skull says "I have seen the future, there are no flags" to which Cap replies "Not my future!" and continues fighting, leading to teleportation of Red Skull to Vormir.

Having the Tesseract for quite some time, and being aware of Greek mythology (familiar with terms like Yggdrasil, the tree of life, alien tech etc) did he somehow manage to use Tesseract to see the future, or what else was the above conversation about?

He did NOT literally see the future, of course, because that isn't at all what happened (in his future). He is simply talking about his vision for the future, presumably when the entire earth is ruled by him and Hydra so there is no need for national flags.

This is just a villain being overconfident in his victory and using metaphorical language.

Also, the space stone (The Tesseract) doesn't give you the ability to see through time. Red skull was never even able to get it to manipulate space, all he ever did was tap it like some sort of energy source for his weapons. The moment he first touched it outside of that he got teleported elsewhere, possibly Vormir.

In Universe: Figure of speech

Out of Universe: Foreshadowing.

He can't see future as he will not purposefully ruin his future by being stonekeeper and nothing suggest he can see the future, he just said for impact but it was also for audience as foreshadowing as spotted by reddit user. And Captain America having no flag on his cloths is also nod to Nomad.