What's the difference between multi label classification and fuzzy classification?
Multi-label classification (Wiki):

Given $K$ classes, find a map $f:X \rightarrow \{0, 1\}^K$.

Fuzzy classification (a good citation is needed!):

Given $K$ classes, find a map $p: X \rightarrow [0, 1]^K$ where $\sum_{k=1}^{K} p(k)=1$.

In multi-label classification, as defined, there is no "resource limit" on classes compared to fuzzy classification. 

For example, a neural network with a softmax layer does fuzzy classification (soft classification). If we only select a class with the highest score, then it will become a single-label classification (hard classification), and if we select top $k$ classes, it will be a multi-label classification (again hard classification).

Fuzzy classification:        [0.5, 0.2, 0.3, 0, 0]
Single-label classification: [1,   0,   0,   0, 0]
Multi-label classification:  [1,   0,   1,   0, 0]


As another example for multi-label classification, we could have $K$ neural networks for $K$ classes with sigmoid outputs, and assign a point to class $k$ if output of network $k$ is higher than 0.5.

Outputs:                     [0.6, 0.1, 0.6, 0.9, 0.2]
Multi-label classification:  [1,   0,   1,   1,     0]


Practical considerations

As demonstrated in the examples, the key difference is the "resource limit" that exists in fuzzy classification but not in multi-label classification. Including the limit (in the first example), or ignoring it (in the second example) depends on the task. For example, in a classification task that has  mutually exclusive labels, we want to include the "resource limit" to impose the "mutually exclusive" assumption on the model.

Note that the $\sum_{k=1}^{K} p(k)=1$ restriction in fuzzy classification is merely a "definition", there is no point in arguing about a definition. We can either propose another classification, or  argue when to use - and when not to use - such classification.