Question: Suppose we have two transformers with same turns ratio, \$N\_2/N\_1\$. First transformer \$A\$ is of \$100:10\$ and the second transformer \$B\$ is \$1000:100\$. Both have same turns ratio so both can step down the voltage by 10 times theoretically. If we assume size of wires are identical at primary and secondary, current and power ratings are same (I am not sure). But eventhough turns ratio is the same for both transformers, the number of turns is different in primary and secondary.

So my question is does the number of turns in turns ratio matter? I have a feeling that \$ B \$ has some advantages. But I am unable to figure out.

Answer: 
Any given core material, cross section and frequency has a maximum volts per turn. If you want to use the winding at a certain voltage, then you need enough turns to support that voltage.

For instance, low frequency mains transformer steel will only run up to a peak field of 1.7 T or so before it saturates. If you had a core that was 10 mm x 20 mm, and wanted to run it at 50 Hz, then the fastest the field swings, for a sine wave voltage, is 2pi.B.f = 6.28x1.7x50 = 534 T/s. In the 200 mm2 of the core, that's a peak flux change rate of 0.1 Weber/s, meaning that core at that frequency will only support 0.1 V/turn.

A 240 Vrms mains winding on that core would need to support 340 Vpeak, so would need 3400 turns minimum.
Transformer design has to optimize a lot of different variables. And I am not an expert. But the transformer primary needs to have enough turns so that it acts like a big inductor and prevents excessive current from flowing. If you take a particular transformer core, you can either wind the primary with many turns of fine wire (high voltage primary) or wind it with fewer turns of thicker wire (lower voltage primary).

But in both cases you MUST fill the whole winding area with copper to make sure the transformer will work well and have low resistance and handle its rated power.

You cannot just put one turn of thin wire. That is basically a short circuit, not a transformer. So, if you choose a transformer core based on power requirements, then choose your operating voltage, then you can calculate how many turns are needed to prevent oversaturation of the core. THEN you choose a wire diameter which will fill the available space efficiently.

That is more or less how the design process works. Once the primary is settled, you choose the number of secondary turns to get the desired secondary voltage. Once you know the number of secondary turns, you choose a wire diameter that fills the space effectively.

Generally, you end up with a lot of turns on your primary for 50 or 60 Hz power transformers.