It is somewhat puzzling:

In[66]:= Assuming[Or[a b >= 0, a b < 0], Solve[a - 1 == 0, a]]

Out[66]= {{a -> 1}}

In[67]:= Assuming[Element[a b, Reals], Solve[a - 1 == 0, a]]

Out[67]= {}

In[68]:= Assuming[Element[a b, Complexes], Solve[a - 1 == 0, a]]

Out[68]= {{a -> 1}}

More puzzles:

In[175]:= Assuming[b == a, Solve[a == 1, a]]

Out[175]= {}

In[176]:= Assuming[b == a && Element[b, Reals], Solve[a == 1, a]]

Out[176]= {}

In[177]:= Assuming[b == a && Element[b, Reals], 
 Solve[a == 1, a, Reals]]

Out[177]= {}

In[178]:= Assuming[b == a, Solve[a == 1, a, Reals]]

Out[178]= {}

In[179]:= Assuming[b == a, Solve[a == 1, a, Integers]]

Out[179]= {{a -> ConditionalExpression[1, b == 1]}}