Holding the company’s equity has the same payoff as a European call option?

As titled, I never got clear as why it is the case?

and also by saying equity shareholders have limited liability, why this indicate it is an european call?

Draw a payoff diagram for a call and this may make sense.

If you buy a stock for say $10 and the company goes bankrupt then you lose your $10 (limited liability). If the company ends up having assets that are worth $100 a share then your payoff is $90 a share.

Contrast this to a call option:

Lets say you buy a call option with a strike price of $10 and for argument sake say this costs $10. If the company goes bankrupt then you dont exercise the option and you lose your premium ($10; limited liability) which is the same as buying the stock. However if the assets turn out to be $100 per share at expiration then your payoff is $90. Same as owning the stock.

Now this relaxes some assumptions but I think it illustrates the point.

Is this from the discussion of the structural model for credit analysis?

If so, you have to understand the assumptions that underlie that model. Given those (unreasonable) assumptions, holding equity is the same as holding a European call option. But it sounds as though you’re trying to say that it’s true in general.

It isn’t.