Aniseikonia is usually much less of an issue with a contact lenses on the other eye after surgery than it is with glasses since the size difference is smaller. However as you discovered it is still possible since people have different sensitivities. Some people can adapt to larger size differences than others. There is no good explanation for "why this is happening" to you other than bad luck that you are perhaps more sensitive than most.
re: "but with 6D IOL plano was achieved, and then, is 9D IOL on my left eye "
The power for an IOL depends on measurements of the eye like the power of the cornea, they measure the size and shape of your eye and use that to calculate an approximate IOL power. It isn't possible to tell what IOL power someone needs just from their contact lens power. Two people could wear the same contact lens power, but need IOL powers that are a few diopters different from each other. It partly depends on how much of your eye's focusing power is in the cornea, and how much is in the natural lens. You also unfortunately can't tell from the power one eye uses what power the other eye will need because it is possible the measurements of the two eyes aren't the same. Surgeons use formulas that take eye measurements and calculate the approximate IOL power needed to achieve certain focal points for the eye. e..g the formula will say that an IOL power X will likely leave the eye focused at Y diopters (whether 0D for perfect distance, -1.75D for a bit nearer for monovision, etc).
If you didn't have Aniseikonia issues when you wore contact lenses pre-surgery, then after your 2nd surgery you aren't likely to have issues either. Likely whatever monovision worked for you with contacts should likely work for you with IOLs. However you'll note I'm not being definite, unfortunately there is guarantee given you are more sensitive to differences than others, but the doctors should be able to make a more educated guess by running the numbers. The doctors should be able to use your eye measurements to be able to calculate the approximate level of aniseikonia between the eyes that you have now, what you would normally have had before cataract surgery, and what you would have after surgery. Since you are having trouble with the issue, it seems worth asking the doctor to explicitly do the calculations. Whether there is an issue will likely depend on how similar your two eyes are, how much of the focusing power of each eye is in the lens (natural or artificial) and how much is in the cornea.
Your interests have not been served by your surgical team failing to inform you of the problems with your eyes working together (aneisometrophia and aniseikonia) after surgery. It is to be expected. Moreover given the amount of your myopia in your LE one would not expect a contact lens to take care of the problem. Please read this carefully: http://www.medhelp.org/user_journals/show/1648102/Consider-ALL-the-Options-Before-Your-Cataract-Surgery-Working-Through-Whats-Best-For-You
also since your RE saw 20/20 I wonder why you had cataract surgery at all and further why your LE which has 20/30 was not done first.
If you wrote you LE RX correctly you have 7.5 diopters of astigmatism and you will need to deal with that by toric IOLs or cornea surgery laser or incisional.
Use the search feature and read the many helpful posts by JodieJ on anisekonia from previous discussions.
Here is a paper on the issue for your doctor:
"Predicting refractive aniseikonia after cataract surgery in anisometropia"
Here is one site with an IOL power calculator that mentions it comes up with aniseikonia if both eye's measurements are used:
"Apart from the original input data on the left side, the results for the 2 IOLs selected are given as tables, stating IOL power, resultant refraction and expected aniseikonia."
What is done is done. Still doesn't make sense to do cataract surgery on the eye with the better vision. In any case I think you are stuck with removing the left lens and putting in an IOL for the intolerable aneisokonia/aneisometrophia. With the comments of Software Developer and my article you should be able to make a better choice. Aiming for a post op refraction of 0.00 (plano) RE and -1.75 LE would be mini-monofocal with distance bias. It works very well for many people. To use your eyes together you would need glasses in a no-line multifocal form. You would be able to do many things without glasses. JCH MD
After getting second opinion and reviewing your comment I had my left eye IOL surgery on Feb 18th. It's been just about 4 days since.
My current right eye is plano with 6.5D IOL, advanced from slightly more from my 2 month followup of -0.25. My doctor and I chose my left IOL targeting -1.75 with 9.5D IOL(SEQ Ref -1.79).
I hoped my unbearable aniseikonia would be almost gone after surgery, but I still experience noticeable aniseikonia which is somewhat better to handle than before the surgery though. I have very good near vision with my left eye, I have very clear vision when I use my iphone or anything closer than 1 foot. But things get unclear soon after 1 1/2 foot, and everything is so fuzzy after. My right eye is seeing well from 3 feet and further(20/20).
With my both eyes open, due to the persisting aniseikonia things are not fused well. Mid-range activities like watching TV is not easy. I see very well with my right eye but I see almost nothing well with my left eye. I don't read big letters of 55 inch TV even from 5 feet apart. It affects all my intermediate and distance vision all together.
I am seeing my doctor in 3 week post op. If the difference of diopters is greater than 2 at the time and still have trouble seeing with both eyes after 3 weeks, what could be my best choice to correct or cope with this problem? My corneal thickness is 530 to 540 and I do not feel laser touch-ups is a good option(my doctor also told me the same even before left eye surgery). How about piggybag IOL or even IOL exchange? I know I need to wait at least 3 weeks after surgery to get my left eye diopter accurately but with moderate aniseikonia even after left eye surgery it makes me worried and feel need of informed. Thank you.
Yes, I will try to relax more and hope my brain adjusts to this new vision. After my first 3 week post-op visit I will update you how my visions are settled. I really appreciate your comment.