Why (neoclassical) economic models have no profits, unlike the real world? Consider a simple neoclassical model of an economy populated by firms with a constant returns to scale production function, using homogeneous labour and capital. Firms are equal, and are price takers in all markets. Thus, firms pay factors their value marginal product. In the long run, any economic profit is wiped out by changes in the price of the good, due to the menace of entry of new firms.

In this setting, no income remains after paying out competitive wages to workers (L) and competitive return to the capital hired for production (K). This is, for firm $i$, 

$$pY_i = L_iw + K_ir $$

Now, in real markets, even in competitive ones, firms do make profits, even if these profits are not abnormal. Part of these profits might be distributed as dividends.

In the trivial economic model however, the ownership of the firm is not explicitly modeled. Firms do not pay dividends. 

Why is this? Why is ownership not relevant? Are profits uninteresting? That is what the capitalistic firms cares about! I'm puzzled.
It seems economic models do not explicitly include ownership of the firm (not of the physical capital, an input used to produce).


Many economic model do have profits. We're just more careful about what we call it. See the definition of "economic profit." When you think of profits, you need to ask yourself if the money you are making is simply a fair exchange for the costs you incur---explicit and implicit costs---or whether some portion of that profit represent money gained above and beyond the (true) costs of production (see "economic rent"). Indeed, this is the only sensible way to think about the world.


  It seems economic models do not explicitly include ownership of the firm


Many economic models do in fact include ownership of the firm. However, most economic models are built for demonstrating some particular idea in relative isolation, as opposed to trying to model every single detail. So if someone wants to demonstrate some effect in a world without economic rents, then a lot of the time modeling the ownership of the firm is less important than modeling the ownership of the capital---and thus it might be excluded. 

If you want to see models that dig deep into the issues of economic rents and the ownership of a firm, I would recommend taking a look at the subfields called "industrial organization" and "corporate finance." (For example, take a look at Jean Tirole's book on corporate finance.


  Meanwhile, in real life, even in competitive markets, firms do make profits on the long run.


This is a bold claim. If you want to be scientific, you're going to have to be precise about how you (1) define profits and (2) define competitive markets, and (3) show (with data and sound analysis) that firms very commonly do make profits as you've defined them. This is how an economist would operate.


  Firms are just an ethereal "thing" that take inputs and produce output with a technology freely available.


Hidden in this comment is an very interesting object of study. What is a firm, really? 
This was an important question (first?) and most famously raised by Ronald Coase. How do we define a firm? Why do they exists? etc. See the "theory of the firm."