Marriage between baptized and sponsor was prohibited in the Latin Church till 1983 and is prohibited in the Eastern Catholic Churches.
The CIC 1917 (Codex iuris canonici, code of canon law) states for the Latin Chuch:
Can. 768. Ex baptismo spiritualem cognationem contrahunt tantum cum baptizato baptizans et patrinus.
Can. 797. Etiam ex valida confirmatione oritur inter confirmatum et patrinum cognatio spiritualis, ex qua patrinus obligatione tenetur confirmatum perpetuo sibi commendatum habendi eiusque christianam educationem curandi.
Can. 1079. Ea tantum spiritualis cognatio matrimonium irritat, de qua in can. 768.
Translation (can. 768 from Eward N. Peters; cann. 798 and 1079 my own, more free):
can. 768 From baptism a spiritual relationship is contracted only between the baptizing, the one baptized, and the sponsor.
can. 798 Also from a valid confirmation comes a spiritual relationship between the confirmed et the sponsor, from wich the obligation of the sponsor stays to hold ever a good relationship to the confirmed and to take care of his christian education.
can. 1079 The spiritual relationship [maybe better: kinship] disturbs only the marriage of the ones named in can. 768.
Can. 1079 is in the chapter about diriment impediments, so a marriage against this canon is void. I intepret it as forbidding only the marriage of baptismal sponsor and baptized, not the marriage of confirmational sponsor and confirmed, but I am not totally sure. This impidiment is seen as of "minor grade" (can. 1042 § 2 n. 4) and can be dispensed from by the Apostolic See (but no other, can. 1040).
This rules were derogated in 1983 when the CIC 1983 came into force. The new code does not know spiritual kinship.
Eastern Catholic Churches
Since 1990 the older canons of the Eastern Catholic Churches are codified into the CCEO (Codex Canonum Ecclesiarum Orientalium, Code of Canons of oriental churches, English here). This code says on this topic:
Can. 811 - § 1. Ex baptismo oritur inter patrinum et baptizatum eiusque parentes cognatio spiritualis, quae matrimonium dirimit.
§ 2. Si iteratur baptismus sub condicione, cognatio spiritualis non oritur, nisi iterum idem patrinus adhibitus est.
Can. 811 § 1. From baptism there arises a spiritual relationship between a sponsor and the baptized person and the parents of the same that invalidates marriage.
§ 2. If a baptism is repeated under condition, a spiritual relationship does not arise, unless the same sponsor was employed for the second ceremony.
Again from this impediment can be dispensed. This time by the local hierarch (can. 795, bishop/eparch).
I would expect the dispense in the Eastern Churches to be given without problems, if there is no wrong in the relationship. But they might look closely for hardly imbalanced or even forced relationships. But I have no empirical basis for this claims.
I know of cases in the Latin Church when some married his/her sponsor. This were cases when the two were in a relationship, but only one Christian at the beginning, the other one came to faith (through the partner), got baptized with his/her partner as the sponsor and the two got married nearly afterwards.