The bread and wine we offer at Mass do become, in Catholic understanding, "truly, really, and substantially, the body and blood together with the soul and divinity of our Lord Jesus Christ"1, and this is what we consume in the Eucharist.
However, this does not mean—as I mentioned in this answer—that the sacrificial offerings are no longer like the originals in any way. As the term "transubstantiation" should indicate, what is changed in the process is the substance, in the older philosophical sense of "that which makes something to be what it is". The accidents, that is, "those things which happen to be properties of an object", do not change.
In particular, that means that one can measure alcoholic content (an accident of the wine) even after transubstantiation; and thus, at least in theory, it would be possible to become drunk from it. Deliberately doing so, of course, would be a gross impertinence to Our Lord. Canon law provides for an excommunication in this or a similar case:
Canon 1367. A person who throws away the consecrated species or takes or retains them for a sacrilegious purpose2 incurs a latae sententiae excommunication reserved to the Apostolic See; moreover, a cleric can be punished with another penalty, not excluding dismissal from the clerical state.
1(Council of Trent, XIIIth session, Canon I)
2Certainly the purpose of becoming drunk off the species of wine would be "a sacrilegious purpose".