Suppose $(p_n)_n$ and $(q_n)$ are two increasing sequences of seminorms on a Fréchet space $X$. Each sequence can be used to define a (complete) metric on $X$. For example,
$$d_p(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \min(1,p_n(x-y))$$ and $$d_q(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \min(1,q_n(x-y))$$
Must these two metrics be strongly equivalent, meaning there exist positive numbers $a$ and $b$ such that $$a d_p(x,y) \leq d_q(x,y) \leq b d_p(x,y) ?$$
No. There are infinite-dimensional vector spaces $V$ carrying two non-equivalent norms $p$ and $q$ such that both $(V,p)$ and $(V,q)$ are Banach spaces. See https://mathoverflow.net/q/184464.
Let $p_n = p, q_n = q$ for all $n$. Then $d_p(x,y) = \min(1,p(x - y)), d_q(x,y) = \min(1,q(x - y))$. These are not strongly equivalent.