R. Yosef Messas has a responsum (Otzar Hamichtavim 3:1825) about late-starting synagogues on Shabbat as it relates to the time for reciting Shema. One point that he makes is that one rabbi said that he heard from many exceptional rabbis that on Shabbat we aren't so punctilious about the time for reciting Shema because the commandment to enjoy Shabbat is biblical while the time for Shema may be only rabbinic.
If we assume that the time limitation is only rabbinic then it is possible that the recitation in the synagogue even after the prescribed time is the real thing (for those who haven't already said it) because it is a fulfillment of the biblical commandment to recite Shema even if it is not a fulfillment of the rabbinic time limit. Thus, it would not be surprising if some congregants engaged in practices that would be indicative of it being the real thing.
Whether you accept the above argument or not, the two practices you mentioned are probably not actually indicative of treating it as the real thing. Regarding kissing the tzitzit, see Shulchan Aruch 24:4 (Rema) that this is for the endearment of the mitzvah of tzitzit, so it may not be so relevant whether the particular recitation is a fulfillment of the commandment of reciting Shema.
Regarding interrupting to join the congregation reciting Shema, this may also not be related to the status of the particular recitation. The law as codified in Shulchan Aruch 65:2 is that one must join the congregation for the first verse of Shema so as not to appear as if he is not accepting the yoke of the kingdom of heaven along with them. This would be an issue whether or not he has already personally fulfilled the commandment of reciting Shema. In fact, the law is actually formulated with reference to someone who has already recited Shema.
As for the bonus question of whether people specifically pray earlier in order to recite Shema within the allotted time, it is certainly possible that some people do it for this reason. In fact, the Talmud (Megillah 23a) states that on Shabbat we come early to the synagogue, and Rashi explains that this is so that we can recite Shema like the Vatikin, which as established in Berachot 9b is right before sunrise.