Is it possible for a person to become "reinfected" with the same strain of a virus?
The nature of infectious agents is that they transmit between organisms. This means that they have reservoirs outside a single host, and hence of course encountering the exact same strain twice is possible if this strain leaves the body and at a later stage is carried back to it. This could typically happen for example with influenza or common cold viruses, spreading from one child to other children at school and later being carried back to the original child through them.

The reason why the child will not develop another illness when being exposed to the same virus again is the adaptive immune response: After the first infection, immune cells specific to a pathogen will proliferate and differentiate both into effector cells to destroy the pathogen, and into memory cells. These can mount a much stronger immune response much faster when they come in contact with the same (or a structurally very similar) pathogen, meaning that a secondary exposure usually does not result in a proper infection and illness. Of course, this mechanism doesn't work properly for anyone with a compromised immune system.

In most cases where a viral disease develops repeatedly, this is due to evasion mechanisms. For example in some viruses, those antigens which tend to be targeted by immune systems have high mutation rates. In this case the "same" virus is actually a mutated strain which causes the same illness, but "looks" different to the immune system, and so the memory cells don't respond. Again, influenza is a great example. The reason why we get flu again winter after winter, is because this virus tends to mutate over the year with relatively low infection prevalence. Then during the winter, when the conditions are favourable for flu infections, typically only a small number of strains cause epidemics - because we were maybe immune to last year's influenza, but this year's has mutated antigens which our immune systems do not recognise yet.