I'm sure there's a certain generic term for this kind of behavior but since I'm a little hard of understanding I don't know what to search for. Generally I want to know exactly what I wrote in the heading.
With periodic functions I mean not only sine like in Fourier-Series but also rectangular functions.
The answer depends on the topology.
Philosophically, in any "nice" topology, a function $f$ can be approximated via convolution with an approximate identity/mollifiers by uniformly continuous functions. Moreover, if $f$ is periodic, then so are the approximants.
Note that if the approximants are uniformly continuous and periodic, they are Bohr/Bochner almost periodic and hence they can be approximated by trigonometric polynomials in $\| \, \|_\infty$.
It follows that if the topology is "nice" and weaker than the topology given by $\| \, \|_\infty$ (which most topologies on functions are), a function can be approximated by periodic functions if and only if it can be approximated by trig polynomials (i.e. almost periodic).
Note that there are many concepts of almost periodicity that are well studied/understood for functions (and even measures): Bohr/Bochner, weak, Stepanov, Weyl, Besicovitch almost periodicity.