Let $M\in C^1(\mathbb R, \mathbb R^{n^2})$. Assume that there exists $a_1, a_2>0$ such that for any $u:\mathbb R\to\mathbb R^n$:
$$ a_1 |u(t)|^2 \le u(t)^T M(t)\cdot u(t) \le a_2 |u(t)|^2. $$
I know that if there exists $a_1>0$ such that $u(t)^T M(t)\cdot u(t)\ge a_1|u(t)|^2$ for any $u:\mathbb R\to\mathbb R^n$, then $M$ is called uniformly positive definite.
But what about this case?
Not a full-fleshed answer, but an help in order to "stratify" the question and examine the first "strata", topological, whereas the second strata looking purely about differentiability.
Consider the case where we don't have functions $t \mapsto u(t)$ but plain vectors $u$ (we drop the (t) everywhere).
For any norm $\| ... \|$ on $\mathbb{R}$, taking square roots :
$$\sqrt{a_1} \|u\| \le \sqrt{u^T M\cdot u} \le \sqrt{a_2} \|u\|$$
(for any $u$), we get what is called a strong equivalence between two norms.
When you have a single inequality as mentionned in the last sentence, see keyword "coercivity".
Knowing that, what are the constraints brought by the $C^1$ context ? This is the second strata, but I am tempted to say devoided of topological aspects.