Trying to assign multiple values to the same Embedded Data field across multiple branches. Doable?

Depends on how many options you have.
The brute force way could be this:

v2 

now has the value

Second Value, First Value

. You could create a branch for each option and append the contact info.

Another approach could be to use Loop & Merge (LM). This takes less time, but give a dirty solution with extra commas:
Step 1: You start with a Pre-LM block, where you ask your question.
Step 2: You create a loop and merge block, selecting loop off a question and the required values in the second field.
Step 3: You use the second field as the question text and use the TRUE-FALSE display logic to not show the question to the respondent.
Step 4: You use piped text to pass all the possible values to another question text. Once again hiding it.
Step 5: You assign the value of this question text to an embedded variable.

Demo here.
Point to not: If you try to set the embedded variable directly skipping step 4. You'll end up with repeated values.