It’s due to so called vowel harmony of Turkic languages. There are technically no end to suffixes! Mind you, suffixes themselves should be harmonized as well.
In your example of “nasılsınız”, as the last vowel of the word nasıl is ı (not i), the following suffix should also contain ı, hence the suffix sınız.
Turkic language learners mostly agree that, once you get the rules, which is very systematic, once you get the hang of it, it begins to flow.
Yes we have something similar in Tamil....in spoken vernacular "ease" (rather than written which does not follow this much along with its spoken register). It happens with some consonants too when speaking....all due to agglutination process and smoothing/harmony need.
There is one South India language that does have explicit vowel harmony in even the written form (Telugu)....as it is overall much more vowel based in its phonology to begin with (splitting off the family branch earlier etc).
In general agglutinative languages (the suffix addition feature you mention) often understand implicit (spoken) or explicit need (spoken + written) for vowel harmony.
In Turkish/Turkic geographical vicinity, you also have uralic (potential linguistic early brethren) like Hungarian and Finnish and mongolic/altaic like Mongolian, Korean and Japanese (also potentially Tungusic like Manchurian).
They all iirc have various vowel harmony norms as consequence of agglutination basis.
Other language families (that create more discrete brakes in morphology i.e are not readily agglutinative) are missing out on this awesomeness heh.
