Why is $Mac_k(m_1||m_2) = ⟨F_k(m_1), F_k(m_1 \oplus \overline{m_2})⟩$ not a secure MAC?
We want to forge the tag for $m = m_1 || m_2$. The tag we need to produce is:

$$\operatorname{Mac}_k(m_1||m_2)=⟨F_k(m_1), F_k(m_1 \oplus \overline{m_2})⟩$$

We'll query the oracle with the message $m_1^\prime || m_2^\prime$, which needs to be different from $m_1||m_2$ to count as forgery. Consider $m_1^\prime || m_2^\prime = (m_1 \oplus \overline{m_2}) || m_2$, which is different from $m_1||m_2$ iff $\overline{m_2} \neq 0$.

This message has the tag:

$$\begin{align}
  \operatorname{Mac_k(m_1^\prime||m_2^\prime)}
  &amp;=⟨F_k(m_1^\prime),F_k(m_1^\prime \oplus \overline{m_2^\prime})⟩ \\
  &amp;=⟨F_k(m_1 \oplus \overline{m_2}),F_k((m_1 \oplus \overline{m_2})\oplus \overline{m_2})⟩ \\
  &amp;=⟨F_k(m_1 \oplus \overline{m_2}),F_k(m_1)⟩
\end{align}$$

swapping the two outputs produces the tag for $m=m_1||m_2$.

Thus we can produce forgeries using only a single query to the MAC oracle.