Answering your questions (which may differ from how I would perform the estimation; your approach is not incorrect, but it is overly complex).

How to segment the population

You almost gave the answer. You segment according to income / net worth level. In other words, segment for example in 4 quartiles and use a different ratio of cars per family for each segment

Number of Taxis and Uber

You need to estimate the capacity of a Taxi/Uber in number of trips; and then the number of trips per day (you can take the previous income-based segmentation and apply a certain number of trips per week for the top2 segments). 

Divide the number of trips per day (you'll need to convert the number from week to day) by 1 taxi capacity (theoretical capacity of a taxi * usage rate) and by the average capacity utilization and you'll get the number of taxis.

# of Taxis = # of daily trips / (daily capacity of a taxi * % usage rate of a taxi) 

# of daily trips = number of people in top2 income segments * average number of trips per week / 7 days of the week (or 6 days)

daily capacity of a taxi = hours a taxi operates / average length of a trip

One alternative… is to consider what % of trips you do in Uber vs. in your own vehicle. To me, it's something like 5%, considering only the top quartile. So, instead of considering the estimation above in terms of number of vehicles, I would consider instead this 5% which refers to additional fuel needs (5% * fuel needs of top quartile segment). 

Now let me digress a bit. When you look at this number (which in my opinion is already an overestimation),  you come to realize this will represent something like 2% of the fuel consumption… IT DOESN'T MOVE THE NEEDLE. So you were going to put a huge amount of effort to estimate something that only represents 2% of the total consumption.  This is something you should avoid.

Instead you could consider the impact light commercial vehicles (the best way would be, once again, consider them as a % of passenger vehicles, e.g. 5-10%, without any segmentation, but acknowledging they probably consume 2-4x the fuel of passenger vehicles as they are heavier and have higher usage rate).

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To solve this estimation, you need to estimate the total fuel consumed (in liters) in Dubai by light vehicles and divide by the capacity (in liters) of an average gas station and adjust by the utilization rate.

Since you would already be using this “capacity estimation” as the first layer of your structure (and this is something that requires significant amount of effort to estimate), I would avoid replicating this structure at lower levels unless really necessary for a meaningful estimation, otherwise you will be taking a very long time. Since your were going to do this for something that would only represent 2-5% of total fuel, I am suggesting to use a more simple route.