You need an actual area
Here is the relevant wording of the Way of the Shadow monk's Shadow Step
At 6th level, you gain the ability to step from one shadow into another. When you are in dim light or darkness, as a bonus action you can teleport up to 60 feet to an unoccupied space you can see that is also in dim light or darkness
Teleporting includes your clothes and equipment
The core of your question is if the monk can create their own area of dim light by using a blanket or clothing they wear or carry. That does not work as their clothes and equipment teleport with them, and therefore also must be in an area of dim light, which they will not be if the overall area is not dim or dark. Some spells and effects (like this one) that teleport you do not explicitly say they teleport you with the gear you wear and carry, but there is a Sage Advice Compendium entry on it:
Misty step doesn’t say the caster can bring worn or carried equipment with them. Are they intended to leave everything, including their clothes, behind? No, the caster’s worn and carried equipment are intended to go
Some teleportation effects do specify that you teleport
with your gear; such specification is an example of a rule
being needlessly fastidious, since no teleportation effect in
the game assumes that you teleport without your clothes,
just as the general movement rules don’t assume that you
drop everything when you walk.
So for the purpose of teleportation, the clothes are treated as part of the character, and the entire character needs to be in dim or dark light. Therefore, you cannot use them to create the shadow conditions you need. You must be in an actual area of darkness or dim light.1
Size of the area of dim light or darkness
The target area is described as an "unoccupied space". Outside of combat, an area small enough for you to physically fit in might be sufficient. In combat situations, space is defined on page 191 PHB:
A creature's space is the area in feet that it effectively controls in combat, not an expression of its physical dimensions.
and a table on that page shows that the space of a Small or Medium creature is at least 5 by 5 feet large. That means you need at least a 5 by 5 feet area of dim light or darkness as the target space to teleport to in combat.2
While the ability does not state that the starting area is a space of dim light or darkness, in any combat situation for practical purposes you will be in such a space, and will have to make a call if that space the character is in counts as dim light or not. If you use a grid, most light sources have a circular range in multiples of 5 feet, and the DMG gives you this rule (page 251):
Choose an intersection of squares or hexes as the point of origin of an area of effect, then follow its rules as normal. If an area of effect is circular and covers at least half a square, it affects that square.
so if more than half of the square is in darkness or dim light, you can teleport from the square. If they are not, you cannot. If you are not using a grid and stick to narrative combat, then the DM will have to tell you if the area you are in is sufficiendly dim or dark to teleport from.
1 There are ways to create the conditions you need in a way you can carry. For example, you can cast darkness (which you conveniently know as a class ability) on an object and carry it around with you, and you can even put it into a bag or other container, to suppress the emanating darkness while you do not want it. The problem with that is that you need to see the space you teleport to, and normally you cannot see through magical darkness, but there are ways around this, too.
2 You can question if an unoccupied space must be the same size as the space controlled the teleporting creature. I think in the absence of other defintions of space, this seems to be the most natural interpretation, as a creature's space and occupied space in combat are the same, and you also will turn that space into an occupied space by teleporting there. Outside of combat, you might get away with a smaller dim or dark space.
P.S. Dim-light tents
This related question is similar, but extends the scope of how to deal with the PCs setting up a small tent and moving inside to trigger dim light or darkness conditions. I think that would be legitimate. In combat, setting up a tent is impractical, so this tactic would be relegated to exploration time.
It mayb e tricky to enable the Shadow Step feature with it, you'd need to set it up just right to block enough light so you can be in dim light but still look out through an opening, as you otherwise would not be able to see the target space. But triggering other features, such as the Twilight cleric's Steps of the Night feature that have no line-of-sight requirements should work fine.