The short answer: No
The long answer:
The Sun only has an analemma (the figure-8 shape traced on the sky as a result of imaging the Sun at the same time (good luck avoiding clouds) each day) because of two factors.
- The Earth is tilted on its axis and that axis remains pointed in the same direction in space (toward the direction of Polaris).
(A more precise explanation is that the axis does precess (or "wobble" like a spinning top) ... but in a very long cycle that takes nearly 26,000 years to complete. So on the scale of single human lifespan ... it doesn't appear to move by any noticeable amount and requires very precise measurements to detect.)
- The Earth's orbit is an ellipse ... not a circle.
While nearly circular ... it isn't perfect. The point along an orbit when an object is nearest to the barycenter of the orbit (in our case... to the Sun) is called the periapsis. But since this periapsis is for an object orbiting the Sun, it gets a special named called the "perihelion". The point of the orbit located farthest from the barycenter is called the apoapsis. Again... it gets a special name because it's an orbit around the Sun so it is called the "aphelion".
But the consequences of the Earth getting nearer to the Sun during half of its orbit ... then getting farther from the Sun during the other half of its orbit ... means Earth's velocity through space changes. As we get nearer to the Sun it is as if we are "falling" so we speed-up (our velocity through space is faster) and as we get farther from the Sun we slow-down (our velocity through space is slower).
Earth needs roughly 365 days (365.24) to complete an orbit. There are 360° in a circle. This means that each ordinary day on Earth (formally known as a "solar day") is 24 hours. In that 24 hours, we've moved forward in our orbit around the Sun by just slightly less than 1°.
It turns out that the amount of time needed for the Earth to spin 360° on it's axis (one "sidereal day") is just slightly less than a solar day. It's roughly 23 hours and 56 minutes (roughly 4 minutes short of a solar day ... that's a roughly rounded value).
So why the difference between a "solar" day vs. a "sidereal" day?
Side note: sidereal translates to mean "of the stars". This comes
from the notion that ancients noticed that the debris left by a
"falling star" (meteorites) that landed on Earth was mostly made of
iron. They surmised that stars must be made out of iron (not realizeing
that meteors are not actual stars that fall from the sky). Their
word for iron is "sidero" (or "sider" pronounced like "cider" ... the juice you
squeeze from apples and possibly ferment into a beverage. It is
pronounced the same. Sidereal is not pronounced like "side-real". It
is pronounced like "cider-eal". Alas, I digress). Sidereal day means the "day of the stars". In other words, if you note the time when a star passes through the meridian (the imaginary north/south line that separates the "west" side of the sky from the "east" side of the sky) ... and then wait to see when that same star passes through the meridian on the following day, the amount of time that has passed will be roughly 23 hours and 56 minutes.... not 24 hours.
Once the Earth has completed one 360° revolution on its axis (one sidereal day) it will also have moved forward in its orbit by nearly 1°. This means that while distant stars would appear in the same point in the sky, the Sun will not... it will be slightly behind the meridian. We will need to let the Earth spin for another 4 minutes until the Sun returns to the same position. But alas... even this has exceptions...
Recall that the Earth speeds up and slows down as it travels through space between it's perihelion and aphelion points. This means the Sun wont precisely be at the same position in the sky... it'll be fractionally ahead or behind depending on if the Earth is speeding up ... or slowing down. This accounts for the left/right variation in the Sun's position in the analemma. The Earth's axial tilt accounts for the up/down variation in the Sun's position. And when you trace out a whole year, you get a shape that resembles a figure-8.
Stars, on the other hand, are much farther away. If you image a star once every 24 hours (assuming no clouds block your view) then each night, the star will shift forward by roughly 1° ... after enough days the Star will be below the horizon and you will no longer be able to image the star. This means the shape you'll get... is that that the star will trace out an long arc ... but after a few months it will no longer be visible at the same time (you would have to observe it at a different time of the day ... and/or it may be lost in the daytime sky depending on the location of Earth in our annual orbit around the Sun.