Theorem: Let $G$ be a connected graph of order 2 or more. Then $G$ is the underlying graph of an irregular multigraph (weighted graph) if and only if $G\neq K_2$
Not sure about proving the 'if' part of the theorem.
An idea is to begin with the connected graph $G$, the pick a degree that is repeated and choose an incident edge so that the other vertex on the edge has a different degree (do I need to prove there will always be such an incident edge?). Increase the weight of that edge until the degrees of both its end-vertices are not found in the whole graph.
Repeat the steps for all vertices that have degree same as another until we have an irregular weighted graph. So such an irregular multigraph (weighted) graph does exist.
If $M$ is a multigraph and all parallel edges between pairs of vertices are replaced by a single edge then the resulting graph is the 'underlying' graph of $M$
A weighted graph is another representation of a multigraph - instead of parallel edges, each edge has a "weight".