How plausible would it be for animals to normalize the birth of monozygotic twins in their reproductive cycles?
As a whole, your question is opinion-based, speculative and not really suitable for Bio-SE.  However, several of the building blocks that you describe have a reasonable biological foundation, which might be useful to know.

For instance:


There seems to be a genetic component To monozygotic twinning (see e.g. ghr.nlm.nih.gov). This means that the trait could, very hypothetically, be selected for, and a population with a higher level of monozygotic (identical) twins could develop (ignoring other possible detrimental effects). Different frequencies of MZ twins is also know from castle breeds and other species.
The level of shared genes is the driving force between kin selection. This is part of the explanation  for the reproductive system in social bees (and other eusocial insects), where (usually) a single queen reproduces and her workers (all sisters), which are 75% related to eachother due to the haplodiplod genetic makeup, all work towards the common "goal" of producing new sister queens (which will then be 75% related to all the workers).
This gives some (again, very hypothetical) basis for your idea that "...they shared the exact same DNA there would be less ingrained concern over which gets to spread its genes".
Monozygotic twins often describe an especially strong sibling bond, which lends support (again, very hypoythetically) to your idea of "...expect them to coordinate efforts in the acquisition of food and evade potential predators.". There is also some research to back this up (e.g. Fortuna et al. 2011. Twin relationships: A comparison across monozygotic twins, dizygotic twins, and nontwin siblings in early childhood), but there are also studies that indicates that monozygotic twins show similar sibling behaviors as other types of sibling pairs (see e.g. Mark et al. 2017. Using Twins to Better Understand Sibling Relationships). This is not a field I know well and there is most certainly many more relevant sources to look at.  


The introduction to Mark et al (2017) contains many references on MZ twins that should be relevant to your "thought experiment", e.g.:


  ...Fraley and Tancredy’s (2012) later work, which suggests that twin children rely more heavily on their co-twin for safety and security than do non-twin siblings.