I'm interested know about some basic metrics that could be used to discriminate trees based on their branch lengths. I know of branch length unaware topological/tree shape-based methods such as Colless and Sackin's indices and am curious whether there are similar metrics that utilize branch length to discriminate. For example, the two trees below have an identical topology but are very different.
There are a variety of measures of tree similarity implemented in the TreeDist R package. (Your phylogenies appear to be from the R ape package, so hopefully you are somewhat familiar with R.)
Other tools include treedist from Joe Felsenstein, which computes Robinson-Foulds and the Branch Score Distance (which sounds closely related to your interests, described in this paper). You can see that software page for a lot more information.
For a practical description of the different tree comparison metrics you can see this article.
I would suggest taking a look at the manual and the multiple vignettes associated with the TreeDist package to learn more about how exactly those metrics work. There appear to be a wide variety of metrics in there that I have never heard of, such as clustering entropy or split entropy or Nye similarity.
This package also includes tools for making visualizations of differences, such as that shown below:
I don't know whether this is actually used in phylogeny, but if you restrict the problem to cases where the topologies are the same, each tree can be described as a point in a n-dimensional space where n is the number of branches, and each axis represents the (possibly normalized) length of one of the branches.
You can then compute distances between points (i.e. trees) in that space.