I'm going to clear up a misconception in your question before we get into 19th century Europe. Comparing conquests of Ancient Persia, Rome, and the Mongols to that time is a bit of a fallacy. Not only did those empires all exist centuries apart from each other already, but they also existed and fell many centuries prior to the 19th. The world had changed significantly between antiquity and modern (ish) Europe, and it would be best to view your question from the reality of Europe at that time. In addition, the Persians, Romans, and Mongols all integrated territory in different ways from each other and depending on the situation. But that's outside the scope of my area and this question.

While all the great powers of 19th century Europe were all trying to find ways to gain advantage over the other powers, including land grabs, the reality was not that simple. Taking control over a province is one thing, but to occupy and annex an entire nation requires a vast amount of resources, money, and manpower. In addition to the drain on your nation, you also have to factor in the opinion of the populace of the annexed country, which may not be, well, fond of being a province of a rival empire that they probably despise. Chances of the populace not cooperating, actively sabotaging, or outright rebelling against your rule would be high, which would drain even more resources, money, and manpower from your nation. You also invite your other rival empires to gleefully fund and support the rebellions in your newly annexed provinces, in order to weaken you further and drain your empire's ability to be a threat to them elsewhere. Ultimately the powers of Europe would see situations like this and decide that annexation of more than a few provinces (at most) was not worth it, preferring either friendly relations with the target country in question or a regime that was cool or hostile to their rivals if possible.

With this being said the Napoleon and the wars surrounding his attempted domination of Europe are the best answer to your question. Napoleon Bonaparte put his siblings on the thrones of Spain and the Netherlands, while also crowning himself King of Italy. Napoleon also founded the Confederation of the Rhine, which contained a number of western German client states that were directly loyal to him. While not outright annexations to France in most cases, the regimes in power were expected to be so loyal to Napoleon that they could be viewed as annexations in all but name. When Napoleon's brother Louis, now the king of the Netherlands, began to ignore Napoleon's orders and try to govern with the interest of his people in mind, Napoleon had him disposed and outright annexed the Netherlands to France. Why hadn't Napoleon annexed these territories in the first place? Well, as previously mentioned, directly administering a territory is expensive, so it was much cheaper for Napoleon to put a person loyal to him on the throne and let them do the ruling and administering of the people. In addition, this method provides a (very thin) veneer of legitimacy that it isn't Napoleon ruling these territories. They're independent, they're just friends with France, that's all, right?

Napoleon's aggressive annexations and power grabs led to the other powers of Europe spending over two decades seeking to impede, limit, or outright destroy his empire if possible. Seven coalition wars were fought against France seeking to contain or restrict its power. The British Empire spent millions of pounds for years with the specific objective of undermining Napoleon in continental Europe (the extent of which could be its own question entirely). The Spanish people fought a long, protracted guerilla war against their new king Joseph Bonaparte, which involved Napoleon committing troops to put down the rebellion for years. Troops he could have been using to fight the other European powers he was at war with. The British sent money, material, and manpower to the Spanish forces to support their cause. The Peninsular War, as it is known, was such a drain on French resources that Napoleon called it "the Spanish ulcer." It is also the conflict where the term "guerilla" originated from, as the Spanish used large scale hit and run tactics to wear down the occupying French armies.

Ultimately, to answer your question, the reason that European powers did not fully annex other European nations in the 19th century is that it tended to be cumbersome, expensive, and impractical to directly rule a foreign country full of people who may hate your occupation. Besides the potential of inviting conflict with other European powers, the massive drain on a state's resources led many nations to conclude it was not worth the trouble. Facilitating friendly relations with the target country or making sure that country was not friendly with your rivals was considered a much better use of money and resources.

Citations:

Black, Jeremy (2008). "British Strategy and the Struggle with France 1793–1815."

Hindley, Meredith (2010). "The Spanish Ulcer: Napoleon, Britain, and the Siege of Cádiz". Humanities. National Endowment for the Humanities. 31 (January/February 2010 Number 1)

Zamoyski, Adam (2008) "Rites of Peace: The Fall of Napoleon and the Congress of Vienna"