If the concentrations of fat are the same in different types of yogurts, it is very likely that the viscosities are not similar due to presence of other deciding factors such as difference in the concentrations of gelatin or (whey) proteins and other components of yogurt and even additives.

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I think the fat concentration does not greatly influence the viscosity of yoghurt. However, the variety of starters, the addition of starch and other additives affect much.

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I think there is relationship between viscosity and fat concentration as it is included in TS so as TS increases viscosity increases...

whey protein, guar gum, gelatin, fibers, etc would greatly affect viscosity with the same fat content

No, the viscosity of milk products like yogurt are independent of fat, if not food additives (thickeners, gelling agents, texture enhancers, etc). Moisture content, and the protein may also affect the viscosity.

If milk is homogenized properly, and for sure fat has been dispersed in equal concentration throughout the gel and well distributed in protein network,, then an impact on the viscosity will be prominent, otherwise it will be separated and come on the surface (particularly larger particles) without having any impact on viscosity. In the latter case only fewer concentration of fat 0.5 to 0.8% having size of less than 0.2 microns will remain within the gel. 

It is quite obvious that yoghurts with same fat contents have equal viscosities. It would be interesting if yoghurt with different fat contents as a variable having all other components in similar concentration particularly proteins shows same viscosity? Through experiments, it is found effective homogenization and distribution of fat trapped in protein network has impact on viscosity. Types of culture among other factors has a greater impact as well.

Fact, that the different fat content in fermented milk products, as in yogurt has effect on the structure and on the viscosity. By my opinion, as the lack of fat results poor, soft texture (lower viscosity) the higher fat content increaseses the viscosity due to the incorporate of the homogenized milk fat globules into the casein network and due to the tixotrophy. In the case of yogurt, the differences may be not too remarkable due to the smaller fat content difference ( 1.5-6.0%). But if You would like to investigate this, bee sure that every other condition (samples have to be prepared from same milk, heating rate, homogenization temperature and pressure, the composition - e.g. protein content, additives -, the type of starter, the inoculation rate with starter, the fermentation temperature, the pH of yogurt, the storage condition before the measure, and the temperature of measurement be same! Only the fat content be different...). 

 I try to attach my own results ..... All the best!