The type of behavior you describe can occur after a fashion. 

During the normal human reproductive cycle a dormant oocyte is stimulated to mature, that is to reenter active meiosis.

(I am assuming that you know most of these terms, but you learn more using the biofundamentals site or the wikipedia!)

The dormant oocyte is arrested in prophase of meiosis I, that is when homologous chromosomes align and crossing-over (genetic recombination) occurs. 

Once activated, the oocyte passes through the first meiotic division, which produces two cells; the secondary oocyte and a much smaller cell, the first polar body. 

In most cases, the first polar body dies. 

The secondary oocyte arrests until it is fertilized by a sperm. 

Fertilization restarts the meiotic process.  The oocyte undergoes a second meiotic division, again producing two cells, the egg and the second polar body.   

The fertilized egg contains two pronuclei, the female pronucleus (the product of the second meiotic division) and the male pronuclues (delivered by the sperm). 

These will fuse to form the zygote, which develops into the new organism. 

For example, the first meiotic division can produce an extremely large (compared to normal) polar body.

This aberrantly large polar body is essentially a second egg, and can be fertilized by a sperm.

So one dormant oocyte can give rise to two zygotes.

So, back to your question (finally).  Are these two zygotes more similar that 'standard' fraternal twins (which are produced from two completely independent eggs)?

The answer is 'no', because the mechanism of meiosis and fertilization. 

You obtain one set of chromosomes from your mother and another from your father.

As a cell enters meiosis, all of the chromosomes replicate -- the replicated chromosomes, known as chromatids, remain attached to one another through a specific structure, the centromere. 

During prophase of meiosis the homologous maternal and paternal chromosomes align with one another  and undergo recombination.   

At the first meiotic division, the paired homologs separate, but the chromatids remain attached to one another through their centromeres.

So the secondary oocyte and the first polar body receive quite different versions of each homologous chromosome.

They are, in fact, more likely to be different than two independently derived secondary oocytes.