Pace u/kobathe I'll have to provide a different answer, especially with respect to the 19th century. Though I would like to say a few things about periods before then as well.

First, the Ottoman Army always had a sizeable portion of Christian soldiers. I mean, always. If there is any point of a state like the Ottoman Empire waging wars against others for political and economic reasons, those are (beyond other, ideological reasons): getting taxes and recruiting soldiers. Now, you can recruit soldiers in three different ways. You can have a central, citizen-army like that of Napoleon, but that was not a thing until the 19th century. Alternatively, you can have a devshirme army. You take non-Muslim children from their families, teach them about Islam and make them learn Turkish (you achieve this through sending them to Turkish families with whom they grow up) and you have a janissary army. But again this is not what matters for us here, because these chaps become Muslim. The third option is what is relevant here; you conquer Serbia, you obviously find a quite complex social system with nobles and all that. What Ottomans did mostly is leaving the local nobles and the local system intact, welcoming but not directly encouraging conversion to Islam or Turkification. If there is a war, then you spread the word and expect your Serbian vassals (among others) to turn up, which they duly deliver mostly. In the Battle of Ankara in 1402, when many Turkish soldiers from the Anatolian principalities deserted the Ottoman Army, the Serbs stood ground and did not surrender until the Sultan himself (that would be poor I. Bayezid the Thunderbolt who was married to Olivera Despina Hatun, the sister of Serbian King Dejan Lazarevic) was captured by the Timur's army. It would be an exaggeration to think these Serbs fought for a shared Ottoman identity (the marital connection and the pressure of a possible Hungarian invasion played a role, İnalcık argues), of course, but it is certainly not true that the Ottoman Army never had Christian soldiers. The same story applies to the conquest of Constantinople in 1453.

I agree with u/Eeate in that the 19th century was an important turning point. After the mid-19th century reforms dubbed tanzimat, you not only no longer have the janissaries (abolished in 1826), but you are now supposed to have a citizen-army including Muslims and non-Muslims alike. In practice, many non-Muslims found a way of circumventing this, by a re-invention of cizye tax, payment of which rendered a non-Muslim exempt from military duties. Yet, this did not apply to each and every member of the Ottoman society. So you have individuals like Kostaki Musurus Pasha, who served as an Ambassador in many places including London (quite famous there during the Russo-Ottoman War of 1877-78 as well!) and Athens (mind you, being an ethnic Greek himself). The translation bureau of the Ottoman Empire was largely made of Greeks until the Greek Independence and then of Armenians until the early 20th century, roughly speaking. Oddest of all for outsiders based on my experience, during the Balkan Wars in 1912, the Minister of Foreign Affairs of the Ottoman Empire was Gabriel Noradunkyan Efendi, an Ottoman Armenian (Findley, p. 239). These examples might easily be enumerated. What I wish to argue is that it is quite unthinkable that none of these individuals serving in the Ottoman Army, in various branches of Ottoman state quite to the best of their ability, did not feel in any sense as Ottomans. I strongly doubt that. This is not to deny that nationalism, separatism, and a mutual feeling of distrust among different ethnic and religious groups in general were on the rise. But the rise clearly did not include every member of non-Muslim communities.

References and Sources

Carter V. Findley, Osmanlı İmparatorluğu'nda Bürokratik Reform: Babıali, 1789-1922 (İstanbul: Tarih Vakfı, 2014). This one is also available in English, first published in 1980 as Bureaucratic Reform in the Ottoman Empire. It involves details on bureaucratic reforms of the 19th century and also provides more information on some non-Muslim members of the bureaucracy.

Halil İnalcık, 'Bayezid I', TDV İslam Ansiklopedisi, vol: 5, (İstanbul: 1992). Available online at: https://islamansiklopedisi.org.tr/bayezid-i

It's a great question and I'm glad you already noted the existence of multiple identities because that is going to be a big part of my answer here.

First, though, let's also ask ourselves the question, "Did Muslims in the Ottoman Empire ever identify as Ottomans?" or more specifically, "Would a Muslim Albanian identify more as an Ottoman than a Christian Albanian would?". Those are also great questions and will have nuanced answers just like answers to your question will be. I can't answer any Albanian questions because I know nothing about Ottoman Albania except that the best Ottoman dictionary was the work of an extraordinary Albanian Şemsettin Sami Bey.

Instead, let's travel across the Ottoman Empire for an answer I am qualified to give, past Istanbul, through Anatolia and into the eastern reaches of modern Turkey to a place that is sometimes known as the Kurdish part of Turkey. I am going to use some examples of Ottoman thought and identity in a small Christian community clustered around the cities of Harput (next to modern Elâzığ), Diyarbekir (modern Diyarbakır) and especially Mardin (no change here). This community was the Syriacs.

Without going into a long, complicated history of the Syriacs, know that their (liturgical) language is Syriac, a modern offshoot of Aramaic. Their church and rites are distinct from those who live around them, although it shares some commonalities with Armenian Christianity and indeed the Ottoman governmental structure, which used the Church to administer its Christians actually organised the Syriacs under the Armenian Patriarchate in Istanbul, but more on that later. Also because identities are complex, the Syriacs are often lumped into a larger Assyrian national identity but this was not primarily how the Syriacs viewed themselves at the time.

The problem for the Syriacs was that they were a small minority in their "homeland"; they were easily outnumbered by Kurds and Armenians and even Turks in many places. Worse both the Armenians and Kurds were an existential threat: the Kurds being often associated with brigandry and other forms of violence, and the Armenians in the form of a threat to the independent Syriac church. There are instances of violence between these two communities over who would own which ancient church or monastery, and "conversions" from the Syriac Church to the Armenian Church were a pain point. On top of that, many Syriacs had adopted Armenian as their native tongue or were intermarrying and losing their "true" identity.

All this lead up to say that Ottomanism - the notion of a cross-denominational Ottoman identity - found fertile ground in this community as a protection against these threats. It's well known that the revolution of 1908 engendered a lot of excitement about liberty, equality, fraternity and justice amongst Ottoman Christians (e.g. Greek or Bulgarian) and we can see this in the writings of Syriac intellectuals of the 1908 to 1914 period. Teaser for my next post: those intellectuals wrote in Ottoman Turkish.