Is there a basic/elementary mass in the universe? When speaking about electrical charges, it seems every particle either has a charge $+1$ or $-1$, in units of the electron charge. Therefore, we have a fundamental charge.
But what about mass? Is there any kind of such mass that every other mass can be seen a sum of those basic masses?
 A: When we come to the elementary constituents of matter, we come to the quantum mechanics regime and the special relativity space time description. In classical physics, masses are conserved and additive. This is not true in the microcosm of atoms, molecules and particles. There masses are the "length" of the special relativity four vector , $(E,p_x,p_y,p_z)$ , and are not an additive quantity and are not conserved. It is energy and momentum that are the conserved quantities. By contrast, charge is an additive conserved number characterizing elementary particles.
In elementary particle studies one has discovered elementary constituents of the proton, for example, which is composed out of three quarks and innumerable internal particle exchanges, which conserve charge and other quantum numbers. The mass of the proton is the "length" of the sum of the fourvectors of the innumerable constituents. 

But what about mass? Is there any kind of such mass that every other mass can be seen as superposition of those basic masses?

This is where experimental and theoretical research are at the moment: it  is a four vector addition that will define the mass of a complex system, not superposition, because mass is not a conserved quantity.
A: Short answer: No. 
Longer answer: Many of the masses in the Standard Model appear to be essentially random numbers. There's no reason to believe that all the non-zero mass particles are integer multiples of any smaller value. 
This isn't experimentally falsifiable, though. For instance, we can't rule out that all masses are integer multiples of $10^{-12}~\rm eV$, since no masses are yet known to that precision.
A: It's a question of scale. And scale matters, this is where physicists take about effective theories where they ignore higher energy effects, or equivalently, short scale effects. 
At the effective level of the atom there appears roughly to be a set of basic masses: the mass of an electron, the mass of a proton. Below this scale the picture becomes murkier and questions arise about even what a particle is. Can we say that something that has a half life on the order of nano-seconds a particle? 
Classically speaking, charge and mass were seen as something that continuously varies; it was an empirical finding to discover that both were discretised. If we look at the classical theories of mechanics and electromagnetism they take charge and mass variables as extensive quantities. It's only with the advent of QM that theoretical arguments for a discrete structure in the small were found. For example, Dirac is famous for discovering a quantisation condition for charge, which famously involved the existence of magnetic monopoles...and which also famously have not been discovered.
