I mean, everybody and their sister were in the First International.

Jokes aside, the International Workingmen's Association of 1864 was a socialist organization - of the 1860s. And its history full of contrasts and internal frictions between the various currents of workers' movements of different orientation and inspiration should serve as evidence enough of its composite nature; also, the development of what we came to identify as late XIX and early XX Century socialism - that is, in short, Marx's or Bernstein's versions of socialism - was parallel and in part subsequent to the experience of the First International.

In its early days, any attempt to adapt it to our very simplified contemporary version of a Communism-Marxism-Socialism trajectory back in time would fail. Since that's way beyond the scope of your question, I will focus on the central issue: the First International was born to collect various national formations of workers' organizations and associations; born in part out of the economical developments (what Marx would call the numerical expansion of the working class) and in part out of the great defeat of the national movements during 1848. It contained and welcomed therefore both elements of "radical" nature (those which would result in Marxism, anarchism, etc.) and "national" elements, contiguous to what a Marxist would define the most progressive-revolutionary strata of the new bourgeoisie. And that, in its early days, almost regardless of their character. The "internationalist" and "materialistic" version of the International experience was far from hegemonic at the time - and in certain regions never actually became hegemonic.

Incidentally, it wasn't just Garibaldi to join the Internazionale but the other Italian founding father (Italy had four of them) Giuseppe Mazzini was a founding member of it - with some distinctions. As Mazzini explained [November 14^th 1864 to F. Campanella]:

The International Association, a good thing in theory, is too much under the influence of a German, Marx, a petty Proudhon, divisive, spiteful, who talks of nothing but war among classes. The English section is good; the others worth little. We keep friendly contacts, but without committing too much to affairs that would only waste time and money of our workers.

It is clear that Mazzini was looking at the "national workers' movement" and its possible collocation within the International; and the accent in his case - as for Garibaldi - was very much on the "national" part. It was Mazzini nonetheless to break more energically with the International Association when in 1871 - following the mounting tension between Marx and Bakunin (who had spent three years in Italy from 1865-67 with the original mission of contrasting the hegemony of Mazzini's ideas) and the experience of the Paris Commune - he took a strong anti-socialist (or better anti-International) stance, even if he focused his polemics for the most part on the rejection of Bakunin's ideas (Mazzini remained unfamiliar with most of Marx's work until his death in 1872).

As for Garibaldi, his participation and sympathetic stance came both from a spontaneous inclination towards certain socialist ideas (that he had after all learned in his early days in Nice - and he remained mostly faithful to those original spontaneistic ideas) and from the need to contrast Mazzini's version of republicanism with his own, especially since his ambiguous collocation within the Italian Monarchy (Garibaldi was not only one of those republicans who had accepted the King but also one who had been involved in more than a few collateral projects produced from the overly conspiratorial mind of King Victor Emmanuel II) made his personal prestige much more significant than any ideological core in retaining his personal basis of support.

On November 13^th 1871 Engels commented on Garibaldi's support, explaining that

He shows more good will than [ideological] clarity [...] nonetheless his last letter [...] is of the greates value to us.

Engels was concerned with the possibility of reaching Garibaldi in his "confinement" of Caprera without his letters being intercepted by the Italian authorities.

This may appear a puzzling state of things for an Italian founding father - to be almost a captive, while receiving a pension from the State for his services, soon to be elevated to the immortal glory of a perpetual national hero. By 1873, Garibaldi - who had spent most of his life living the life of a soldier, an exile, or a fugitive - was also beginning to show the first signs of phisical decline; and for a man that had based his popularity on his actions rather than his though, this also meant that his political influence was growing more limited. From Caprera Garibaldi could not control the Italian republicans, radicals, anarchists or socialists: among his followers, many continued their contacts and participation within the International, in positions either alternative or contiguous to those of Bakunin - and usually against Mazzini's republicans. But those "extreme" forces of the 1870s could not coalesce in a substantial political factor of influence; and progressively the Italian "socialist" movement shifted towards the composite platform of early XX Century - a blend of ideological marxist orthodoxy, reforms, and an inclination for currentism.

Garibaldi and Mazzini were - by the time Italian Socialism took its more familiar face - dead, but not forgotten. When, around 1890, the Italian Socialists choose to celebrate some Marx related anniversary (I think it was 1893 and the ten years of his death - but can't remember for sure) by distributing a bust of the German socialist, they choose to send one to Engels, thinking he would be pleased.

Sadly, they Italian artist had "put too much of Garibaldi in it", according to the old Engels.

There is a marked tendency in the years after the Italian Risorgimento to empty the actual political and social experience of their substance, and leave them as symbols: names of battles, fortresses, dead patriots great and small. Atop of all those: the four founding fathers: Garibaldi, Mazzini, Cavour and Vittorio Emanuele II - men who disliked each other, who disagreed on many substantial issues, who argued, at times violently, and - if we believe certain rumors - attempted to have each other killed.

Long before Fascism, not only accurate historiography but reality itself was sacrificed to the altar of national unity. Can we really blame them? It's a big lie but it's for the better. For the nation's own good.

But onto the other side of your argument: if being a member of the International or of any socialist organization was a reason in itself to warrant a damnatio memoriae, then I suppose Mussolini would have had to add another layer to his bronze face.

Garibaldi and Mazzini were celebrated by the Regime for what they were in context: that is - if we believe fascist historiography - precursors of Fascism, founders of the Italian nation, men who had sacrificed a certain ideological apparatus to the need of their time. With Garibaldi this transformation was a bit harder than with Mazzini, whose ideas and "religion of the nation" offered a better chance of deformation - Garibaldi the individualist, unruly, red shirted man running for his life from a score of persecutors... But realistically, how to destroy all those busts?

Someone may even argue that Garibaldi - the true man, not the symbol - would be irreducible to Fascism; but the symbol was manageable. He had after all submitted to the King as a living man; he could submit to the Duce as a ghost.

And finally - from 1945 Italy was a democratic country: you could keep a picture of Stalin or one of Roosevelt, one of the Pope or one of Marx, or both. Historiography of the Risorgimento has not always been of the highest quality, even since then. But I will be seriously afraid the day someone consents to erasing Garibaldi from Italian history.