The Shadow Step feature is not magical
The Sage Advice Compendium (page 17) answers the question on determining a game feature as magical as below:
Determining whether a game feature is magical is straightforward. Ask yourself these questions about the feature:
- Is it a magic item?
- Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
- Is it a spell attack?
- Is it fueled by the use of spell slots?
- Does its description say it’s magical?
If your answer to any of those questions is yes, the feature
The Way of Shadow monk's Shadow Step feature (PHB, p. 80) is defined as below:
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. You then have advantage on the first melee attack you make before the end of the turn.
The feature is not described as magical in nature, so it does not match the 5th criterion. Also, while there is a spell called "Teleport", the lack of italics or other explicit reference to this spell in the description of Shadow Step suggests that this does not qualify for the 2nd criterion either.
Seeing as it meets none of the above requirements, Shadow Step is not a magical feature.
In relation to the changes to magic in Waterdeep: Dungeon of the Mad Mage, I expect you are referring to the "Alterations to Magic" section below (emphasis mine):
Halaster doesn’t make it easy for creatures to enter or leave his dungeon. No spell other than wish can be used to enter Undermountain, leave it, or transport oneself from one level to another. Astral projection, teleport, plane shift, word of recall, and similar spells cast for these reasons simply fail, as do effects that banish a creature to another plane of existence. These restrictions apply to magic items and artifacts that have properties that transport or banish creatures to other planes as well. Magic that allows transit to the Border Ethereal, such as the etherealness spell, is the exception to this rule. A creature that enters the Border Ethereal from Undermountain is pulled back into the dungeon upon leaving that plane.
From the above, it looks like the only intention is to stop the player from moving in and out of the dungeon or between levels. As Shadow Step is dependent on line of sight, it seems unlikely that this feature could be used to circumvent the intentions of the Alterations to Magic, and other forms of movement which are explicitly magical are not affected outside of trying to escape or move levels. As a result, it does not seem likely that Shadow Step would be directly affected by Alterations to Magic.