What should I use as an alternative to the combination of Soy oil and Sesame oil (1 teaspoon each) when I am frying fish in a pan?
I need at least one ingredient that can replace the Soy and Sesame oils for frying the fish in the pan.
You need to consider the role of the oils in the dish.
The soy oil is almost certainly used for its frying properties (to help prevent sticking, and more efficiently transmit heat from the pan to the food).
For this function, you can use any vegetable oil or shortening for frying, based on your dietary preferences and your relative's sensitivities. Grapeseed oil is an excellent choice with neutral flavor, and a high smoke point. Canola is also a good choice, as is refined peanut oil, and many others. Pick one that your relative isn't allergic to.
If animal fat is acceptable, frying in clarified butter (or ghee) is very delicious.
Sesame oil has a strong (and in my opinion, wonderfully delicious) flavor and aroma, and is used for that, not for its frying characteristics. You can leave it out, although it will change the flavor of the dish. There isn't really a substitute.
If your relative is allergic to all oils and you don't want to use butter or another animal fat, you could steam-griddle the fish, rather than frying it. Its a different technique, and will have a different texture and flavor (much less browning). Similarly, you could try en papillote, which requires no oil, or poaching--again no oil. It is very difficult to fry without oil or fat of some kind (perhaps, by definition, impossible since without the fat, you aren't frying but just dry griddling), but choosing another technique friendly to fish may avoid the problem.
Use any type of heat resistant fat in place of the oils. Any fat used will impart it's taste on the fish, so avoid lard and other animal fats. Traditionally fish is fried in canola, olive or sunflower seed oil.