The persons soul is what they are, essentially they're essence.
In the end we are what we believe others see because it is from this point of view that we analyze ourselves without emotion because we are using someone elses perspective.
to have a second language merely helps a person to express themselves in a different way
To discuss whether "to have a second language is to have a second soul", we first have to look at what is a "soul". As Vicky has mentioned, the word "soul" is an ambiguous term to an extent that every culture has a different definition for what a soul is. Therefore, it is very difficult to be fully agreed or disagreed on the topic. For the purpose of this discussion, I think it is safe to assume that the soul the quotation mentions is in accordance to the Western definition of a soul -- a soul is that which give body life -- since the quotation is said by Charlemagne, the emperor of the Carolingian Empire.
To a greater extent, I believe that it is not enough to simply define "soul". We also have to come to an agreement upon what does it mean "to have a secondary". I think that to have a second language means not only to be able to understand a language other than the mother tongue, but also to be able to understand the culture linked to that language.
As language is the path of communication, possessing two languages extend a person’s range of expression. A personal example would be that I frequently encounter problems with translating Chinese into or from English. Even though I speak and understand both languages quite well, there are certain barriers that one language cannot cross. So, in a way, my body is separated into two different ways to reason, or two different “souls”.