The question at first appears to be about (what is in some jurisdictions) a legally defined relationship, but then proceeds to say that it is not 'about legal recognition of the term'. These two elements of the question are at odds with each other.
If one were wondering whether wife would be the correct term to use for this purpose in a legal context, one would have to seek the answer in the law of the particular jurisdiction; the question would then not be about English language in general.
On the other hand, if the question is whether one could use the term in informal social interaction, then it is not clear how the legal technicalities of common-law marriage are relevant to one's choice of the term. When one introduces that person to one's friends, the friends won't care in the least how one's involvement with that person may affect one's welfare entitlements, or how one would split the property with that person in case of separation. The fact that the person is one's common-law spouse is unlikely to be noteworthy in any context other than the legal ones.
One can thus, in an informal social setting, refer to one's common-law wife by any of the terms that one would use for a person who is not one's common-law wife, but with whom one is in a relationship that is similar for social purposes, such as girlfriend, partner, long-term companion, or significant other. And, yes, one can refer to that person as one's wife, if one wishes to convey that one regards the person as one's wife, and that one expects one's friends to treat her accordingly.
In so far as there is a connection between one's being in a common-law marriage and what terms one uses in informal social settings, it, in fact, goes the other way round from what the question assumes. While there is no reason why the former would influence the latter, one's consistently referring to a person as one's wife (as has been pointed out by Mary in a comment) may be a factor in determining whether one is in a common-law marriage with that person (a part of the definition of a common-law marriage is that the two people hold themselves out as married).
It should be noted that this question has very limited application, as very few jurisdictions nowadays still use the traditional concept of the common-law marriage. There are places where the law includes some rules that in some way resemble it, but don't do so fully, and do it under a different name.