Why didn’t the monarch let the Empire of India merge into the British Empire?

The answer to this question is also partially an answer to your other question. Essentially, the British wanted to keep up the appearance of being a spiritual successor to the old Mughal Empire which had ruled much of India for two centuries. The British thought that it would be more palatable to the vast Indian population - which was dramatically larger than Britain's own population - if they felt like they had their 'own' empire which was not directly ruled by the British Parliament and government, but instead a viceroy, who intentionally emulated the highly dignified, imperial style of the Mughal rulers. This appealed more to the Indians, who preferred the Mughal style of ruling to the more 'modern', mercantile nature of the British East India Company. By presenting British rule not as a totally new and foreign thing, but merely as a successor to the Mughals, they maintained more prestige and somewhat more acceptance among the Indians.

The British were smart with how they ruled; they kept as many Indian princes and lords in positions of power as possible, to serve as intermediaries between the viceroy and the population. This preserved the appearance of Indians still ruling the country, and in many places this was actually true; Indian princes sometimes had very large amounts of autonomy, and the only thing that was different was that they passed on the tax revenue to the viceroy and the British administration rather than the nawab and the Mughal administration. In a way it actually was an "Indian Empire", since the population was overwhelmingly Indian, as well as many local lords and rulers, and the majority of the army. All that was different was that a tiny minority of people at the top were British. For India to be 'just another part' of the rest of the British Empire would have been to give holding possession of India less prestige, and also upset the Indians more, since they wouldn't feel like they had an important and unique part in the British Empire (which the British made sure they felt like they did, especially in the 20th century).

Bonus question: Why wasn‘t the British monarch‘s title Emperor, since he was ruling over the British Empire, but rather “King of the United Kingdom (at some point Ireland) and the Dominions of the British Commonwealth“ plus Emperor of India. Isn‘t that odd since the title of Emperor is of higher rank than a King‘s yet the monarch was ruling an empire?

The British were - and in many respects still are - a people that highly valued their history and traditions, and historically Britain had always had kings, ever since the end of Roman rule. They had a king, and always had. They did not need or want him to be anything more than that. Additionally, in the 18th and 19th centuries, the term 'emperor' had somewhat negative connotations in Britain; the highly conservative and authoritarian governments in France, Austria, Germany and Russia all had emperors, and they were all absolute monarchs, or at the very least had significantly more power than their British counterpart. To British eyes, being an emperor (or its native translations such as kaiser and tsar) meant being an absolute monarch that ruled without a parliament. This was obviously totally unacceptable to the British people, and they weren't even willing to just change the name for formality's sake. When Queen Victoria suggested this it was instantly shut down by her government and parliament. They gave her the title of Empress of India as a salve, since, as explained above, the Indians were accepting of being ruled by an emperor/empress but the British weren't. This was for historical reasons; the Mughals had always ruled as emperors, but the British monarchs never had.