Before we get into seasons, life, etc., we have to address the question of whether or not such a configuration could exist. At first glance, I see no reason why two planets couldn't orbit a star in the same way Janus and Epimetheus orbit Saturn. It could be a bit tricky if the planets are gas giants, because they would have a strong gravitational influence on one another, but if they were terrestrial planets, they would be fine. Such a configuration could exist.
The more important question here is whether or not such a configuration could form in the first place. From the excellent site you mentioned,
Janus and Epimetheus may have formed from a disruption of a single parent to form co-orbital satellites. If this is the case, the disruption must have happened early in the history of the satellite system.
Wikipedia, interestingly enough, says the exact same thing:
Janus and Epimetheus may have formed from a disruption of a single parent to form co-orbital satellites, but if this is the case the disruption must have happened early in the history of the satellite system.
Let's consider how these planets could have formed:
- Collision of two protoplanets. This is the idea behind the Giant Impact Hypothesis, which says that a protoplanet collided with Earth early in the history of the solar system; the Moon was formed from the resulting debris. The problem is, if this happened between two Earth-sized protoplanets, the result might not be two planets - it could be a planet and a large moon, and some debris.
- The Roche limit and tidal forces. There are a number of questions here that investigate what would happen if certain things passed inside the Roche limit of another body. A body of more than two solar masses could definitely venture inside the Sun's Roche limit, and be torn apart. It's possible that the material would coalesce into two planets; not necessarily likely, but possible.
Both these scenarios could very well result in two co-orbital planets, though perhaps not in the desired arrangement. We'll know more about these kind of ideas if we ever learn about the true source of Janus and Epimetheus. For now, we can say that such a configuration con very well have formed, either from one of these two scenarios or something completely different.
So, to answer
first could planets going around a sun have a similar relationship
Yes.
Next up, we turn to life. It seems clear that the two bodies would be in some way related to one another, and would most likely be made up of the same materials. In fact, in lieu of another impact on one, the two might develop nearly identically, for a while. At least for a few hundred million years.
I see no reason why life couldn't exist on one or both of these worlds. Remember, they're only near each other for a short amount of time, so they don't have a huge influence on each other - most of the time. Going back to your site,
This exchange happens about once every four years.
Maybe this period is different for our planets, but still, you get the idea.
The really interesting thing here is that, because conditions might be really similar on both planets, we could see life develop similarly on both. Sure, there are a lot of factors that influence life's development, so we shouldn't expect the same creatures to develop on both, but life would likely form from the same compounds, and have the same evolutionary history, for a few hundred million years.
how might that affect any life on them?
Life could form, and it might be similar on both.
Finally, we go to seasons. Janus and Epimetheus are both in synchronous rotation with Saturn, so only one side faces Saturn at a time. This means that they are tidally locked with Saturn. I wrote a brief answer on seasons on tidally locked planets, but the setup was different, and it may not be relevant here because tidal locking is not necessarily the outcome.
If there isn't tidal locking, there should be normal seasons if the axis of each planet is inclined with respect to the star. If it isn't, there won't be any seasons, unless the orbit is somehow highly elliptical (See this question and this question). There also won't be a huge amount of interaction between the two planets, so any axial effects should be negligible.
Would it change the seasons much?
No.
