I have seen some people having problem with city merging for various reasons, so here I try to explain it in a quick way.

Question 1: What are the prerequest to merge city?
Answer: You need the medieval era technology Military Architecture before you can merge cities. Also the cities territories need to be adjacenent to each other, similar to how you attach outposts, in fact merging cities is very similar to attaching outposts.

Question 2: How do you merge cities?

Answer: It is very similar to how you attach outposts, you enter/select the city you want to merge another city to and then move the screen over the name of the city you want to merge into your selected city and you should see the merge icon with an influence cost. If you don't see the icon, that mean you have not fufilled the prerequests at question 1.

Question 3: What determine the cost of attaching cities?
Answer: Right now the exact formula is not know but from experience it is based upon several things. First and most importantly is the cost increase for any infrastructure one of the cities lack, thus you should make sure the cities have the exact same infrastructures before merging them. Secondly the number of territories of the target and third the number of districts of the target.

Question 4: What happens once you merge cities?
Answer: It is very similar to what happen when you attach an outpost to a city, the target city is removed from the game, its city center become an administrative center, all territories it had become territories for the city it was merged into, its population join the city it was merged into, the districts/quarters become districts/quarters to the new city and any infrastructure in the now merged city is simply lost but its districts and quarters benefit from the infrastructure of its new city.

Question 5: Why merge cities?

Answer: While merging cities come at a huge influence cost and there are advantages of having many cities, there are some advantages of merging cities. First is you get rid of a city, thus reduce your city count, giving you room for a new city. Secondly it is cheap in terms of industry and money to build districts in new cities, making it attractive to found many small cities, but you need many districts to make things like infrastructure worthwhile and certain emblematic districts get very powerful in large cities which eventually encourage you to merge the cities together. A strategy is to use colony plans to quickly create cities with many districts but no infrastructure to keep the merge cost as low as possible, once you feel the small cities have been developed enough, you can merge them together and thus quickly create a huge city that may reach well over 100 districts.

Attaching is a lot cheaper if the district is adjacent to the city.

Merging cities allows you to create local builds faster but does lose infrastructure buildings (1 less fountain, 1 less aqueduct)

I have found the formula for the cost for attaching Outposts. It is:

30 + 50 * (Outposts attached to THIS city) * (Outposts attached to ALL cities)

Obviously further modified by cost reductions etc.

(Also updated it here: https://humankind.fandom.com/wiki/Outpost)