Effect of the type of oil on a vinaigrette's texture?
Add egg yolk or lecithin to your mixture and whisk it again.

Here is some theory:

To make an emulsion (consistent, not separating oil–water mixture), we rely on emulsifiers – long organic molecules that on one end bind with water, and another with oil. They play role of a glue that does not let molecules of water and oil separate.

The most common emulsifier is a lecithin – compound found in egg yolks.
There is a more complete list of them I found on this website:

Emulsifiers in food