Note: I started this answer by writing up all sorts of wiggle room about the word "day" not intending to come up with an answer. Then I came up with a crazy explanation (rogue planet). And finally and then hit on a plausible one (variable star) that doesn't rely on any of that. I've reordered the answer to put the best answer first, and left the rest in because it's interesting.
Variable Star: Same Star, Different Day
A variable star is a star whose brightness and other attributes fluctuate. These fluctuations can be regular, or irregular. They can be big, or they can be small. They can be caused by changes in the star itself, or by things blocking the star like a dust cloud or swarm of comets.
A planet around an irregular variable star which has both internal and external causes would seem to see a different star every day. The changes in the star itself, plus the various occluding objects in closer orbits around the star, would make it seem "random".
For example, what if our planet had a gas giant in a closer orbit. That gas giant has trapped debris in large clouds at its L4 and L5 points.
These clouds act like giant filters giving the star a different color, intensity, and apparent size in addition to the variations in its actual size. As the gas giant and our planet rotate relative to one another, one field or another (or none) will partially occlude the star. Throw in more than one interior gas giant, each with their own debris field, and you have a heavenly kaleidoscope!
The debris fields are diffuse enough, the variability slow enough to never see an abrupt change. The planet's rotation is fast enough to never have enough time in a day to observe a slow change. To a layman the star sets while fuzzy, small, and red (filtered through a debris field), and when it rises it's sharp and blue (it's fully passed the debris field).
On top of that, there will be normal changes in the appearance of the star as it travels across the sky due to its light passing through different parts and thicknesses of the atmosphere, just as our own Sun looks different at sunrise and sunset and in different parts of the world in different weather. This further complicates anyone figuring out they're looking at the same star without careful observation and consideration.
Until they figure out the very complicated and unstable orbital mechanics of their system, it will appear to them as if they look at a different star every day. Until someone can coordinate tracking their star's changes over a full rotation, nobody can prove it's the same star changing its appearance. This could be a key plot element of the story, people's observations are converging on the conclusion that it's been one star all along, and that threatens some political power based on the orthodoxy of a new star every day.
This configuration does not have to be stable, if it lasts for 10,000 years that will be long enough for it to be "normal" for all of a planetary society's history.
Sidereal vs Synodic Day
There's some issue with the notion of a "day". On Earth there's the sidereal day, and then there's the synodic day. We're used to them being basically the same, but what if they weren't? What sort of culture would grow up around that and what notion would they have for "day"?
The sidereal day is what an astronomer might use for a day. It's the time it takes for the planet to make one full 360° rotation. A planet bound observer can measure this by tracking the angle of the (relatively) fixed distant stars.
Then there's the synodic or solar day, that's the time for a point on the planet to come around and face its star again. It's sunrise to sunrise. That's what your average person would call a day.
They're not the same on the Earth. Because the Earth rotates in the same direction as it orbits (as do most planets) it has to go a little bit further to line up with the Sun again. A synodic day is 24 hours, but a sidereal day is about 4 minutes shorter.
This is the wiggle room in the requirement. Since the question talks about "having a different sun every day" this is the synodic day: sunrise to sunrise.
Who Cares What Happens On The Other Side of the World?
Furthermore, a low tech society wouldn't be able to communicate or travel to the other side of the world. Their notion of "day" would be what they see in their sky. Whether there's another star shining in the sky in the other side of the world would be of no consequence to their notion of a "day".
I Wish There Was More Time in the Day
And then there's the length of the day. We're used to a "day" being a very small fraction of the year, but there's nothing that says it has to be. For example, because Mercury rotates so slowly, and it's so close to the Sun, its solar day is two of Mercury's years! It's tidally locked to the Sun.
Lacking any significant axial tilt, Mercury doesn't have any seasons; there's little difference to Mercury whether it's on one side of its orbit or another. A culture growing up on a Mercury-like planet which is in a star's habitable zone wouldn't have much use for a "day" or "year" as a measure of time.
All this adds up to some more wiggle room to make it work.
This is a variation on the novel A Deepness in the Sky which I highly recommend.
What if we had a rogue planet with life that has adapted to the relatively brief periods it's warmed by a star? What if the reader is not made privy to this, and the characters just talk about a "day" as if it's perfectly normal?
There's nothing that says life, even intelligent life, has to have a metabolic rate as high as Earth life. This life has evolved to very carefully husband and store energy. For most of the time life hibernates underground, nearer to the well insulated planetary core, warmed by the left over heat from its gravitational formation, and the decay of radioactive elements. As it drifts near a planet, a new "day" begins, the surface becomes briefly habitable, and life pours out onto the surface to take advantage of the increased energy available.
This surface explosion of life could take dozens, hundreds, or thousands of years depending on how close the planet has drifted to a star and how large that star is. This doesn't necessarily have to be in the habitable zone for liquid water, imagine the relative bonanza Pluto has compared to a rogue planet in deep space.
Let's say this planet has a very, very slow rotation. As they near a star it might come above the horizon looking a bit bigger than the rest. As they drift closer and closer, parallax will make it appear to move through the sky, getting a little larger. As the planet warms, life would pour out onto the surface for the new "day", their metabolism rising to make the most of the available energy.
At "noon", their closest approach to the star, it would appear largest in the sky, then start to shrink as the planet moved away. Their metabolism would slow and they'd start storing energy for hibernation as the planet cools. "Sunset" would be when they can no longer distinguish their star from any other in the sky and they'd return to their hibernation.
The next time they come to the surface, it's a new "day" around a new star!
You can even play with the planet's rotation and some temporary chaotic orbits to come up with even crazier days.