Is a Catholic "blessing of a marriage" not a wedding?

The issues with a Catholic marrying a non-baptized person in civil ceremony:

1. Such a marriage according to Can. 1086 §1. is invalid.
2. And according to Can. 1086 §2., a person is not to be dispensed from this impediment unless the conditions mentioned in cann. ⇒ 1125 and ⇒ 1126 have been fulfilled.

It is not clear from the post whether these issues were resolved prior to the couple's priest performing a "blessing" of the marriage, and the post is not clear what this "blessing" was exactly that was termed "not a wedding."

Answering

Q1. What, exactly, is the nature of this blessing? How does it affect the status or validity of the marriage?

A1. The nature of this "blessing" is unclear from the post. The first place to start to obtain answers is from the couple's priest who performed this "blessing."

Without knowing the nature of this blessing, the second question is currently unanswerable.

"Blessing of a marriage", as explained by a canon lawyer and the staff of a Catholic parish, is a fairly standard laymen's term which can be used for one of two processes recognized by Catholic canon law as allowing a marriage to be recognized as valid in the Catholic Church.

The Church's Code of Canon Law has a whole section on marriage (canons 1055–1165). Among the regulations it lays down are some having to do with how and where the marriage may take place (the form of the marriage), and some having to do with who may and who may not be married. Under certain conditions, the bishop of the diocese to which Catholics belong may waive most of these regulations in a specific case; but if this waiver (called a dispensation) is not obtained, the marriage is considered invalid by the Church, either because of impediment (a "who may and may not marry" rule was broken), or because of defect of form (a "how and where" rule was broken).

This marriage certainly appears to have been considered invalid by the Church (otherwise the priest would not have felt the need to have it "blessed"). Why?

The Church requires that Catholics marry in their own parish (unless they receive a dispensation to marry elsewhere):

Canon 1115. Marriages are to be celebrated in a parish where either of the contracting parties has a domicile, quasidomicile, or month long residence or, if it concerns transients, in the parish where they actually reside. With the permission of the proper ordinary or proper pastor, marriages can be celebrated elsewhere.

Thus, a marriage such as this—in a state ceremony (at a registry office, town hall, etc.) without a dispensation, which appears not to have been obtained in this case—has a defect of form.

In addition, Catholics require special permission to marry non-Catholics:

Canon 1086 §1. A marriage between two persons, one of whom has been baptized in the Catholic Church or received into it and has not defected from it by a formal act and the other of whom is not baptized, is invalid.

This too is a canon which can be dispensed; but again it appears that no dispensation was obtained in this case. Thus the marriage, as well as suffering from defect of form, also suffers from an impediment. Or rather, it originally suffered from it. Since the man was baptised into the Catholic Church, the impediment "ceased" (in technical terms) before the marriage was blessed.

Thus the original marriage was invalid because it suffered from both defect of form and impediment. Later, the couple wants it to be recognized in the Church's eyes. What can be done about that?

The Church has two processes that can be used, as I stated, to restore the marriage to one considered valid, and both of these are typically referred to as "blessing the marriage". It appears that one of these is what was done by the priest.

Simple convalidation is a procedure done when there was no defect of form in the marriage, only an impediment (which has since ceased, or been dispensed). As canon lawyer Cathy Caridi explains:

Speaking generally, this would likely be the method used if (a) the impediment had become a matter of public knowledge; or (b) one or both spouses had known from the get-go that the impediment existed, and had for whatever reason managed to conceal it.

In this case, if only one spouse knew of the impediment, he or she would simply renew consent to the marriage (as one does with wedding vows) in private, without needing to inform the other spouse. If both spouses knew, but no one else, the spouses would each renew consent to the other (effectively re-exchanging wedding vows) in private, before the priest, and that would be it. If the impediment has become public knowledge, then canon law would require public exchange of consent "in canonical form"—effectively, a new marriage ceremony:

Canon 1158 §1. If the impediment is public, both parties must renew the consent in canonical form, without prejudice to the prescript of can. 1127, §2. [i.e. subject to any specific rules laid down by the bishop]

§2. If the impediment cannot be proven, it is sufficient that the party conscious of the impediment renews the consent privately and in secret, provided that the other perseveres in the consent offered; if the impediment is known to both parties, both are to renew the consent.

In all these cases, simple convalidation results in the marriage being considered valid only from the time when the marriage was convalidated, not from the original wedding.

But that doesn't seem to have been the case here: there was not only an impediment to the marriage, but also a defect of form. In that kind of case a different sort of procedure is called for, also described informally as "blessing the marriage", but technically known as radical sanation.

Radical sanation would require the priest to request a dispensation (ex post facto) from the bishop for the state ceremony, and another for the marriage to a non-Catholic. If these were granted, the priest would ask the couple to declare their vows to each other once more, and then the marriage would be considered valid—not just from that point, but retroactively from the original ceremony.

AthanasiusOfAlex has posted an answer to part of the second question, but because of the already great length of this answer I would prefer to post that as a second response.