To give this a bit more shape, and not run amok of Renaissance historians, I will keep this focused on the circumstances in Italy between the French Revolution (1789) and the completion of the Risorgimento in 1871. Also, this is very general so I may need to expand on this.

Upper class and more educated Italians (pre-unification) were very aware of the cultural similarities across the peninsula that came with a (relatively) common language and the legacy of the Italian Renaissance in particular. This shared cultural and historical heritage was largely remembered in the northern cities where such culture had flourished: Genoa, Venice, Bologna, Sienna. This is often referred to as the italianistá- a byword for shared Italian history and culture: Donatello, Machiavelli, Da Vinci, etc.

So following the events of the French Revolution and Napoleons Empire (1814) the Congress of Vienna was called to stabilize the geopolitical situation in Europe. The goal was to contain any runaway power - like the French had been - and give all major powers on the continent equal footing to prevent another cascade of wars. In those deliberations it was decided that the Veneto and Lombardy (Venice and Milan) would go to the Hapsburg Empire, along with more or less total political control over smaller "independent" polities in the North. The two independent powers on the peninsula were the kingdom of Sardinia Piedmont in the north (led by house Savoy) and the Kingdom of the Two Sicilies (named such in spite of there being only one Sicily) in the South (led by the French Bourbon dynasty). Rome and a large portion of central Italy was of course held by the Papal States. I should also mention that the Congress of Vienna also sought to contain the liberal (i.e. republican) sentiment as well and preserve European conservative monarchy.

After Napoleon's occupation and client state, there were many Italian thinkers who began to imagine a unified Italy. The foremost was Giuseppe Mazzini (1805-1872): a politician and journalist who began to write and call for Italian unification. Mazzini and his supporters - including a young Giuseppe Garibaldi (1807-1882) - were predominantly liberals who believed that a republic would be superior to a monarchy. There are countless others who could be mentioned among those supporting Italian unification but let's just summarize with two important details about most of them: educated and predominantly republican. To offer some example of a more cultural variety (all this politics is a bit dry) Giuseppe Verdi's (1813-1901) opera Nubocco imagines the plight of Italians suffering under Austrian occupation as Hebrews languishing under the oppression of king Nebuchadnezzar in the Bible. They were a passionate group.

Now to your point on foreign powers not being desirous of a unified Italy, it depends on the power. British liberals generally supported the cries of Italian people to unify, but the Austrians who were occupying most of northern Italy this movement threatened Austrian interests. Klemens von Metternich (1773-1859) referred to Italy as "a geographical construct", meaning that there was no political union to unify the peninsula, only the land itself. So to put a finer point on the answer here: with an oppressive force occupying Italy, and an increased sense of cultural connection why didn't Mazzini and his other supporters just work to fight against the Austrians and other conservatives on the peninsula?

Well, they did in 1848. Mazzini and his supporters seized Rome and declared the Roman Republic. Fighting in Milan and the Veneto against the Austrians was fierce but ultimately these revolts were unsuccessful. French emperor Napoleon III sent French troops to Rome to oust Mazzini and almost all of the conflict of 1848 yielded little new change. Napoleon III was very interested in the preservation of the Papacy as a formal geopolitical power and thus against the republican visions of Italy that Mazzini and his supporters held. Ultimately, this led many of Mazzini's supporters either into Austrian prisons or exile, where they continued to lobby for Italian unification under a republic.

Solely because it could take a great deal of time to explain, I will omit the more commonly known details of the Piedmontese rise to prominence in Europe (though I can explain if you would like) and will just summarize most of their goals. Firstly, Sardinia Piedmonte was a monarchy, and thus interested in the preservation of that institution. Secondly it was allied to France, in no small part due to Camillo Benso (1810-1861), the count of Cavour. Essentially Cavour had to balance Napoleon III's interest in the preservation of Rome as a Catholic capital and independent power, the preservation of the Piedmontese monarchy, and the growing calls for unification (many of which were republican).

Early in 1861, Giuseppe Garibaldi took 1,000 irregular volunteers and invaded the Kingdom of the Two Silicies in the south to help bring about Italian unification. Though Cavour and others knew about Garibaldi's plans, they were not the one's sanctioning it. Garibaldi overthrew the Bourbon's regime and shortly after a detachment of Piedmontese troops led by King Victor Emanuel II (1820-1878) of Sardinia-Piedmont were given the kingdom by Garibaldi. THIS is the one take-away, Garibaldi ultimately chose a unified Italy as opposed to an Italian republic.

TLDR; To put a finer point on all this, ultimately Italians shared the cultural identity that you mention (at least among those who were wealthy and urban) and there were in fact foreign powers interested in stopping a popular unification of Italy. The primary inhibitor of unification was that conservative monarchists both outside and within Italy did not want any unified Italian government to be republican as many of the more vocal nationalists wanted it to be. These internal political tensions worked with those foreign tensions to suppress nationalists like Mazzini and Garibaldi and it ultimately led to Garibaldi compromising with the only power that may have been able to unify Italy - the Piedmontese monarchy.

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Sources:

Secondary:

Multiple volumes from Dennis Mack Smith: Cavour and Garibaldi, 1860, The making of Italy, 1796-1870, Mazzini, Modern Italy

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Lucy Riall: Garibaldi: Invention of a Hero

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Raymond Grew: Sterner Plan for Italian Unification

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There are more I could list but these are the ones that stuck with me the most on the relevant topic.

Edit: Some silly formatting stuff.