**Delete reason:** Working on fixing the answer.

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**Tl;dr** 

A graph with this property (let's call it property P) cannot be locally finite, that is, must have vertices of infinite degree (for an example of such a graph, see the answer of Florian Lehner).

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The idea is to apply [Zorn's lemma](https://en.wikipedia.org/wiki/Zorn%27s_lemma), and for that, we define a partial order on the set of all coverings:

$$C\ge\bar C\quad:\Longleftrightarrow\quad \mathrm m(C)\subset\mathrm m(\bar C).$$

Property P basically states that there is no maximal element. Assuming that $G$ is locally finite, I will show the contrary via Zorn's lemma. Let $\mathfrak C$ be a chain. The tricky part is to construct an upper bound to that chain (note: we cannot just intersect all the $C\in\mathfrak C$, they might be disjoint).

If we assume that $E$ is countable, we can choose an enumeration $E=\{e_1,e_2,e_3,...\}$, and define the initial segments

$$E_i:=\{e_1,...,e_i\}.$$

I need the following term

> **Definition.** Given a chain $\mathfrak C$, a subset $\mathfrak D\subseteq \mathfrak C$ is called *end-dense*, if for any $C\in \mathfrak C$ there is a $D\in \mathfrak D$ with $D \ge C$.

Note, that being end-dense is transitive.

We now recursively define a decreasing sequence of chains $\mathfrak C = \mathfrak C_0\supseteq \mathfrak C_1 \supseteq\cdots$, so that each $\mathfrak C_j$ is end-dense in $\mathfrak C_{j-1}$.
Note that every covering $C\in \mathfrak C_{j-1}$ induces a covering on $E_j$, and there can be only finitely many distinct coverings on $E_j$.
Consequently, we can choose an end-dense $\mathfrak C_j\subseteq \mathfrak C_{j-1}$ so that all $C\in \mathfrak C_j$ induce the same covering $\smash{\bar C_j}$ on $E_j$.

We then have $\bar C_1\subseteq \bar C_2\subseteq \bar C_3\subseteq \cdots$ and we can define $$\bar C := \bigcup_i \bar C_i.$$

I claim, that this is an upper bound for $\mathfrak C$: if a vertex $v$ is multiply covered by $\smash{\bar C}$, then so it is in some $\smash{\bar C_j}$. This $\bar C_j$ is induced by the infinitely many converings in $\mathfrak C_j\subseteq \mathfrak C$, hence $v\in\mathrm m(C),C\in \mathfrak C_j$. Since $\mathfrak C_j$ is end-dense in $\mathfrak C$, we obtain that $v$ is multiply covered by all $C\in \mathfrak C$.

Consequently, $\bar C$ is an upper bound for $\mathfrak C$, and Zorn's lemma establishes the existence of a maximal element in contradiction to your property P.