Why don't we get a shock touching neutral wire?

Do not touch even the neutral wire in a live circuit! There are numerous failure modes that could make you dead wrong about not getting shocked.

The neutral wire does have current going through it. However, we do not get shocked when we touch something with current going through it, we get shocked when current goes through us. In this case all of the current that enters one end of the section of wire we are touching also leaves the other end. None goes through us so we don’t get shocked.

Why does no current go through us? From the perspective of a circuit we are, to a first approximation, just a big human shaped resistor. The current through a resistor is proportional to the voltage across it. Since the neutral is at the same voltage as the ground, both our head and our feet are at 0 V. So the voltage across us is 0 V and therefore the current is 0 A.

Why don't we? Actually we could!

Your first line is the assumption which will kill you. Literally.

Neutral wire has a V same as ground i.e almost 0.

In a perfect world, with perfect zero-impedance electrical conductors everywhere, there is no voltage between neutral and ground. In that perfect world, you could safely touch the neutral wire.

Surprise - we don't live in a perfect world...

Electricity transmission doesn't actually have a neutral. Instead it has 3 wires which all have AC on them, where the AC waves on each are out of step with the others. When the electricity gets to your local substation near your house, it goes through a transformer to step down the voltage, and also to kind of "add" the three AC signals. Because the AC voltage waves are out of step with each other, adding them together like this gives a "zero" voltage in the middle, which is the neutral.

To keep the neutral close to ground, it's wired to ground (a physical metal rod into the earth!) through some resistance. It can't be tied to ground too well though, because lightning strikes nearby would go back up the neutral and be incredibly dangerous. (They still can, by the way, but the damage is mostly fairly low because of this.)

Separately, your house also has a physical metal rod into the earth which is connected to the ground points of all your outlets.

So how can this go wrong...?

Well, "ground" here isn't "ground" elsewhere. That's why your house has its own metal rod into the earth. Various things (lightning, chemical reactions in the soil, friction of air over the ground, air humidity, just for examples) make for voltage differences between places. Famously, that's why cows can be killed by a nearby lightning strike - there is enough voltage between each foot, across the ground they're standing on, to electrocute them. So even if the neutral was at "ground" at the transformer, it might not be by the time it reaches your house.

Or connections can get broken. If your house earth becomes corroded or breaks, or the same thing happens at the transformer, all bets are off on whether you're at ground or not.

Or you can get a short-circuit from live to neutral. In theory this should trip a breaker. But suppose you don't have a breaker, or someone has replaced a fuse with a nail in an old house. The resistances along live and back along neutral will be roughly equal, so the neutral wire at your outlet will be around half the live voltage. (And then the wire insulation tends to burn and set the house on fire, but that's a separate problem.)

Or you can get a fault in electricity transmission where one phase connection is broken. Instead of the three phases adding together to cancel out and produce the neutral, now you've got a damn great AC voltage on the neutral. Usually there is protection against this getting out, so all power should be shut off if this happens, but in a rural setting with older kit (or on somewhere like a farm where it's managed by the farmer) I would not bet my life on it.

Or more ironically, something somewhere else might have a fault where it's shorted live to earth. You might be safe to touch neutral (apart from the issues above), but if you held onto earth in one hand and neutral in the other, it'd be the earth which kills you.

Just a few examples there. I'm not pretending this is an exhaustive list of the ways you can kill yourself with the neutral - I'm just trying to give you a flavour of why this is a bad idea, and why your incorrect assumption proves fatal to so many people every year. Don't add to those statistics.

if we touch the wire don't we to become a part of the circuit?

"Circuit" comes from the same Latin word as "circle," or "circus." A circuit is a loop. The only way you can ever have a steady flow of electric current is if it flows around a loop. If you touch a wire, but you are insulated from everything else, then you aren't part of any loop, and there can be no steady flow through your body. But...

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...It gets a little more complicated when we're talking about Alternating Current (AC), because then we can talk about current "flowing through" a capacitor even despite the fact that the two terminals of the capacitor are electrically insulated from each other.

Any time some electrically conductive thing comes near some other electrically conductive thing, there is weak capacitance between them. Notably, there always is weak capacitance between yourself and Earth.

Even if you are well insulated from everything else, you still will feel a strong "tingle" if you touch the "hot" wire in a circuit because a tiny amount of AC current is able to flow in a loop from the generating station, through the hot wire, through your body, through the capacitance between your body and Earth, and through the Earth back to the Earth wire at the generating station.

You aren't supposed to feel any tingle* if you touch the neutral wire, because the neutral wire should be connected to Earth someplace not too far from where you are standing. Effectively, both ends of the "capacitor" are connected to the same thing (Earth), and so it's not really part of any loop or circuit.

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* But note what other answers already have told you. What is supposed to be is not always the same as what actually is. Touching the neutral still is risky because a fault in the wiring could cause a supposed-to-be-neutral wire to actually be "hot."

Don't go around touching live circuits.

Neutral is near to 0 v, lets say 15v in a domestic installation. As all wires has some low resistance (lets say 5 ohm as example), we can model this system as a point of low voltage (15v) with a low resistor (5 ohm) connected from it to real ground (0v).

When someone (don't do that at home) touches the neutral wire, the change in the circuit is that appears a new connection path from the touch point, across the body, to the ground. The body has an (usual) impedance bigger to 1000 ohm (it depends of person, fingers pressure, conditions, shoes, ...).

Now, the circuit is a parallel of connection of two "resistors" from neutral touch point to ground: the wire (5 ohms) and the body (1000 ohms). By circuits law, the amount of current that will go across the body will be 5/(1000+5)=0.5% of the total current. Usually, that means a level of current that can not cause big injuries.

However, these values are too near to dangerous current intensities and, in some scenarios, a really dangerous current can go across the body.

Don't do that at home.

(a comment about biology: if you must touch a wire -don't do it-, touch it with the reverse of the fingers. In case of shock, muscles tends to close the fingers, so, the fingers will jump out of the cable contact. Instead, if you touch with the internal side of the fingers, you risk that shock closes the fingers around the cable, making it a deadly and insurmountable closure).