Not really. Turkish has a feature that is called "vowel harmony". You match suffixes you add to a word based on a category system: low pitch vs high pitch vowels where a,ı,o,u are low pitch and e,i,ö,ü are high pitch.

Ö and ü were already borrowed from German alphabet. Umlaut-added variants of 'ö' and 'ü' have a similar effect on 'o' and 'u' respectively: they bring a back vowel to front. See: https://en.wikipedia.org/wiki/Vowel . Similarly removing the dots bring them back.

Turkish already had i sound and its back variant which is a schwa-like sound: https://en.wikipedia.org/wiki/Close_back_unrounded_vowel . It has the same relation in IPA as 'ö' has to 'o' and 'ü' has to 'u'. Since the makers of the Turkish variant of Latin Alphabet had the rare chance of making a regular pronunciation system with the state of the language and since removing the dots had the effect of making a front vowel a back vowel, they simply copied this feature from ö and ü to i:

Just remove the dots to make it a back vowel! Now we have ı.

When comes to capitalization, ö becomes Ö, ü becomes Ü. So it is just logical to make the capital of i İ and the lowercase of I ı.