Electronic – How do they produce batteries with the same voltage but different current

Current depends on Voltage". So, if the voltage is high, current would be high. Agreed; (I=V/R)

True, if you're asking about resistance.

But, you're asking about a (non-ideal) voltage source - a battery.

The voltage to current relationship of a battery depends on the chemistry, temperature, etc. Cells and batteries are not resistors.

Now, it is the case that a first approximation of a battery is an ideal voltage source in series with an ideal resistor, as another answer points out, but it must be kept in mind that this effective resistance depends on a number of factors. See, for example this.

If two different batteries(with same voltage) delivers different currents, how can we say that they are both 2V batteries?

You've answered your question yourself. They have the same open-circuit voltage.

Clearly, two identical cells connected in parallel provide the same open-circuit voltage as one of the cells but two cells in parallel can provide twice the short-circuit current of the single cell.

There seems to be some temptation here on EESE to assume "plates" refer to or equal cells. I believe this is mistaken.

Cutaway images of automotive 12v lead acid batteries I am finding seem to show that each nominal cell consists not of a single pair of plates, but rather of a stack of many interleaved plates, alternately connected to the cell's positive or negative terminal. In effect, each cell is a number of cells in parallel, though sharing the same bath of electrolyte. Six of these cells in series then form the 12v battery.

Likely, the 11 or 13 plate designs are two currently or recently common tradeoffs in the parallelism vs. size or thickness. I'd expect we'd need a battery designer to explain the advantages or disadvantages of each.

Edit: in fact this is the case. Automative-, motorcyle- etc style batteries often have right on their data plate the number of plates per cell, and 11 is a very common number.