In general, in a Boolean algebra it is not true that if equal terms appear on both sides of an equivalence, then we may "cancel" them (cancellation law). In other words,
\begin{align}
A+C = B + C \,\not\!\!\!\implies A = B
\end{align}
Indeed, consider the case where $A = 1 = C$ and $B = 0$. Then, $A + C = 1 + 1 = 1 = 0 + 1 = 1 = B$ but $A = 1 \neq 0 = B$.
You have the phenomenon in set theory, and essentially for the same reason. To see the analogy, consider the boolean $+$ (the logical "or") as the set union $\cup$. We have have that:
\begin{align}
A \cup C = B \cup C \, \not\!\!\!\implies A = B
\end{align}
Indeed, take $A = \{a\}$, $B = \{b\}$ and $C = \{a,b\}$, with $a \neq b$. Then, $A \cup C = \{a,b\} = B \cup C$ but $A = \{a\} \neq \{b\} = B$.
The deep reason is that boolean $+$, as well as set union $\cup$, is idempotent, differently than arithmetical $+$ in natural numbers. Idempotence means that $a + a = a$ for any $a$ in your domain.
This is what happens with any $a \in \{0,1\}$ and boolean $+$, as well as with any set and set union $\cup$.
Idempotence makes impossible to define a "minus operation" that is the inverse of the idempotent $+$.