Classify doesn't work when the targets are lists, so you need to encode the labels. However, there must be a nicer built-in way to do label encoding than this, but I can't find it:

n = 4;
t = Table[
   r = RandomInteger[1, {n}];
   d = Thread[
     Range[n] -> Map[FromCharacterCode, Range[97, 97 + n - 1]]];
   {r, (Flatten@Position[r, 1] /. d)}, {100}];
ut = Union[t[[All, 2]]];
en = Thread[ut -> Range[Length[ut]]];
de = Reverse /@ en;
x = t[[All, 1]];
y = t[[All, 2]] /. en;
c = Classify[x -> y];


In[2]:= c[{1, 0, 0, 1}] /. de
Out[2]= {"a", "d"}

Since Classify accepts associations, I would use their functionality to encode the classes. First, group your data by class

at = KeySort@GroupBy[t, Last -> First];

then set up the encoding and the reverse lookup for later

ena = Association@MapIndexed[#1 -> #2[[1]]&, Keys@at]
dea = AssociationMap[Reverse]@ena

The actual encoding is then straightforward

endata = KeyMap[ena]@at;
c = Classify@endata

Unfortunately, I do not know of a built-in way to do the reverse lookup, although your code works as is with associations, e.g.

In[211]:= c[{1, 0, 0, 1}] /. dea
Out[211]= {"a", "d"}

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