Does dusk really remain for a shorter period of time at the equator? It is said that the dusk remains for shorter time at equator than the poles. Because, the equator rotates faster than poles. But it is also true that time is the same in every latitude, and if it's true, then the dusk should remain the same at equator as the poles. So, does dusk really remain for a shorter period of time at the equator?
 A: It is faster because the sun takes a higher trajectory through the sky typically, and crosses the horizon steeper and thus faster.
A: "Dusk" is defined as "the darker phases of twilight" (in the evening), so it may be ambiguous. There are in fact 3 different twilights:

which are defined by how far the sun is below the horizon (hence the answer from @tobalt). Since the East-West speed of the sun in the sky is identical across the planet (though it does change over the year), the more perpendicular the celestial equator is to the horizon, the faster the sun goes down.
A good resource for such questions is: https://www.timeanddate.com, which shows the length of daylight (sun above horizon), civil twilight, nautical twilight, astronomical twilight, and night for each day of the year for any significant city.
Here's today's look at Quito ($\phi = -0.18^{\circ}$) and Utqiaġvik, AK ($\phi = 71.17^{\circ}$):


You can see that Quito's twilight is nearly constant over the year, are shorter than Utqiaġvik, where the length of twilight today is zero. In the winter it will be 10 hours long, though that does comprise both dusk and dawn, with no sunrise/sunset between them.
The formulae use to make these figures are involved, e.g. https://gml.noaa.gov/grad/solcalc/solareqns.PDF or https://en.wikipedia.org/wiki/Sunrise_equation .
A: The line that separates day and night (illuminated vs dark side of the Earth) is called the shadow terminator. Now, because we don't experience a sudden lights-on/lights-off transition, but a gradual shift towards nighttime (or dawn, at the other end), you can imagine there's a transitional region - a band of sorts - attached to the shadow terminator, where we experience twilight as we pass through it. A "twilight zone", if you will.

But it is also true that time is same in every latitude and if it's true, then the dusk should remain same as the poles.

Think of the circle that a stationary person or a place describes as the Earth rotates. They complete the full circle in 24h. As this "twilight band" has basically the same width everywhere, it will take up a larger part of the circle at higher latitudes (because the circles get smaller), which means people there will spend more time in the band. The situation is further complicated by the fact that the Earth's axis of rotation is tilted.

The rotation axis of the Earth is at an angle with respect to the ecliptic plane (Earth's orbital plane), and it maintains this orientation in space as the Earth travels around the Sun (Milankovitch cycles aside).

That means that the day-night line (the shadow terminator line) does not pass through the poles throughout most of the year.

For example, when it's winter in the northern hemisphere, the north pole faces away from the Sun for months. So, at the poles, the Sun can dip below the horizon for 6 months continuously, but about half (or more) of that time is some degree of twilight (for details, see this).
A: I get the sense that the OP might have been wondering about what seems like a riddle: if people near the equator are moving faster (due to the earth's rotational speed), shouldn't they "go through" the twilight region faster than people near the poles? But how can this be true since the people at the poles and the equator both keep the same time, and "twilight" is at a particular time?
Several people have given good answers on the geometry and science of why this is, but I wanted to add something that might help the OP's intuition about solving the "riddling" nature of the question itself.
While geometry determines the boundaries of dusk and twilight, nobody has discussed the part of the OP about the spinning of the earth, which (for the sake of discussion) we sometimes characterize as saying that the earth rotates at about 1000 mph at the equator and about 0 mph at the poles.
What nobody has pointed out, @Debanjan Biswas, is that the linear speed of the earth's rotation at your particular location on earth cannot be used by itself to calculate the passage of time. As your own question observes, that speed is very different for people at the equator versus those near the poles.
What you can use to measure time is your angular speed: every location on earth turns 15 degrees every hour, so that after 24 hours, it is pointing back where it was originally. This is true ten feet from the equator, but also ten feet from either pole. It is even true if you sit down right on top of one of the poles and stay there for 24 hours, staring straight ahead. In those 24 hours, your view will stare into space in one direction and then circle like an extremely slow lighthouse, until you are facing the same part of space again 24 hours later.
The reason that this isn't the only factor contributing to the length of periods like dusk and dawn has to do with the angle of the earth's axis, and others have covered that very well. As they have explained, whether dusk is longer or shorter at the equator when compared to one of the poles depends on the time of year. There is a time of dawn and a time of dusk at the equator every day of the year, and those periods of time do not vary much over the course of the year, as the periods of daytime and nighttime are balanced at the equator all year.
As you move away from the equator, however, periods of daytime become longer in the summer and shorter in the winter, and nighttime becomes shorter in the summer and longer in the winter. Move far enough from the equator (to the polar circles) and you find the areas where there are days or weeks with no nighttime (in the summer) or daytime (in the winter) for days or weeks on end. If it's daytime all day, you might have a day-long dusk (if you are right on the arctic or antarctic circle) or no dusk at all (on the poles).
