What does it mean to shed blood in the name of Christ?

and that nobody can be saved, no matter how much he has given away in alms and even if he has shed blood in the name of Christ, unless he has persevered in the bosom and unity of the Catholic Church.

Its a paraphrase of 1 Corinthians 13:3 where "love" (or "charity") is replaced by "must be Catholic":

And though I bestow all my goods to feed the poor, and though I give my body to be burned, and have not charity, it profiteth me nothing. (1 Corinthians 13:3)

That combined with the fact that early proponents of martyrdom basically taught that martyrs go straight to heaven (e.g. the baptism of blood concept), and its obvious its a reference to martyrdom.

Thus the real problem is not that this quote calls for violence (which I don't think it does) but rather that it has replaced charity with a rigid legalism (quite literally)!

Edit:

No, I wouldn't consider my answer the "official Catholic answer," as I doubt there is some official document that officially interprets this official document, but I think its informed by up-to-date information on Vatican II Catholicism.

As to modern Catholic references to the "shedding of blood in Christ's name", Benedict apologized for the crusades a few years back (source), so you're not going to find any modern Catholic saying (from the hierarchy) praising the shedding of blood in the sense of other peoples' blood. You'll probably not find martyrdom being referenced under those exact terms either, but you will find it being praised. For example, from this news article:

The strength to give one's life comes from "profound and intimate union with Christ," Benedict XVI clarified, "because martyrdom and the vocation to martyrdom are not the result of human effort, but the response to an initiative and a call from God, they are a gift of his grace, which makes one capable of offering one's life for love of Christ and of the Church, and thus of the world."

And so far from agreeing with the quote from Florence, modern popes, like John Paul II, praise Protestant martyrs.

I do see at least one instance of the phrase "shedding of blood" used to speak of martyrdom. In the encyclical letter Invicti Athletae, published by Pope Pius XII in 1957, he notes:

There is always a bit of martyrdom in [Christian] virtue. ... not only by shedding of blood is the witness of our faith given to God, but also by courageous and constant resistance to the lure of evil...

The Latin original from which this was translated reads

Cui quidem virtuti, si reapse volumus ad christianae vitae perfectionem cotidie magis contendere, semper aliquid martyrii inest; quandoquidem non solum profuso sanguine fidei nostrae testimonium Deo praebetur, ...

The phrase "non solum profuso sanguine" could be more literally translated as "not only by means of blood poured forth", which leaves the question open of whose blood is being poured forth by whom.

On the other hand, there are uses like that of Pope John Paul II's 1995 encyclical Redemptoris missio:

Without an openness to the Absolute, what does man become? The answer to this question is found in the experience of every individual, but it is also written in the history of humanity with the blood shed in the name of ideologies ...

In this instance, the Latin phrase translated "blood shed" is "sanguine effuso", meaning "blood poured out". I haven't done enough research to tell whether this is a distinction universally used by the Vatican; and I have no access, unfortunately, to the Latin text of the documents of the Florentine Council, to see what their original wording was.

(emphases added)

In modern language 'shed blood' usually means someone else's, but language has changed a lot in 500 years. (read Shakespeare or the KJV).

The sense of the passage is clearly "no matter what good works you do, if you are outside the Church that counts for nothing". It is clearly building up a list of greater and greater deeds - fasting, alms, works of piety - and so literary sense would be that the last item would be the biggest sacrifice of all, conveying the sense that even if you give absolutely as much as you possibly can, still it avails you nothing.

The biggest sacrifice of all would, of course, be giving of your own life for Christ, and I would understand that this is the sense that "shed blood in the name of Christ" is meant. It means to shed your own blood.