There are those that kiss their Tzitzis during Shema at Shacharis as per Kitzur Shulchan Aruch 17:7 and the Kaf HaChaim 18 and there are those that do not. What is the source and reason for those that do not kiss their Tzitzis during Shema at Shacharis?
Ma'ase Rav Siman 39 says that the GR"A would not kiss his Tzitzit during Keriyat Shema. The Kovetz Mepharshim (printed on the side in the Weinrab edition) supposes that he did so because he held that when saying the third paragraph of Keriyat Shema one should gaze at his tzitziyot (see biur hagra O"H on Siman 24 seif katan 8), and kissing them breaks his focus.
I have not the exact halachic sources but read that some poskim considered those kissings during the shema as hefsek, for example : The Vilna Gaon, Chazon Ish, The Steipler, Rav Moshe Feinstein (read there : https://halachafortodaycom.blogspot.com/2012/04/q-page-1.html)
Also I read once that some don't even grasp their tzitzios during shema like for example Rav Chaim Soloveitchik but that's an other subject.
A practice appeared which was to touch the Tzitzis when saying והיה לכם לציצית וראיתם אותו. This is mentioned in the Tur Orach Chaim at the end of siman 61. Previously, in siman 24, he mentioned a Bal Haitur that explained the practice of people to gather together the strings and put them on their eyes, and rationalized, perhaps they do this because of וראיתם אותו and their tzitzis were worn under their clothes.
The Beis Yosef explains all of this in the light that this practice was originally met with much opposition. The Avudraham quoted Rav Natrunai Gaon said it is haughty to do anything publicly which one is not required to do. The Beis Yosef adds now that there are a small amount of purple that do it, it is no longer haughty.
But the B.Y. goes on to quote the Orchos Chaim who brings Rav Natrunai Gaon, but also brought Rav Moshe Gaon who said the same and explained even further once someone thought about his tzitzis while pouring them on, why would he hold them in his hands afterwards? And according to this, he should hold his tephilin when he mentions them. And if you think he should do this, then when he mentions mezuzah he should go put his hand on that, therefore it is improper. He ends off quoting the Eshkol to look at the tzitzis which the B.Y. takes as meaning he agrees with. The B.Y. points to aforementioned Tur in siman 61 which seems to rule accordingly. And also different between tzitzis and mezuzah by explain tzitzis are right there next to you. The B.Y. goes on to quote many Rishonim who say simply looking at tzitzis is a mitzvah in it's own right which would give credence towards taking care to look at them when mentioning them in Krias Shma.
With all of this we understand the words in the Shulchan Aruch that SOME look at the tzitzis and touch then to their eyes and its good to do and all for the love of the mitzvah. The Ramma adds that some people also kiss the tzitzis for this same reason. The parenthesis erroneously attribute this idea as a quote from the B.Y. It is not. The Ramma wrote in the Darkei Moshe that he has seen people doing this and explained that it is also because of love for the mitzvah.
One more point, as of yet no one has mentioned a multiple kissing process when saying the word tzitzis. From the Ramma and Darkei Moshe it is clear that a single instance of kissing them occurred when reaching the words וראיתם אותו. However the Kitzur Shulchan Aruch mentions a practice to kiss them with every mention of the word. The Kaf Hachaim also brings from Chesed L'Avraham who mentions this practice as well.
The Bet Yosef says (paraphrase, see comments) 'some say it is pride to kiss the Tzitzis while praying Shema. And some say there is no pride because it is a very common practice. And the second one is the Minhag.' I guess some people are still Hoshesh to the Isur of Yuhura.