The issue is not just whether something is or is not a prayer, but also whether it is a דבר שבקדושה.
The Beit Yosef brings (OC 565:5) in the name of the Rashba that an individual (not in a minyan) shouldn't say the 13 Attributes in the context of בקשת רחמים (requesting mercy), since that is a דבר שבקדושה (learned from the gemara on Rosh Hashanah 17b). However, when said with trop1, the 13 Attributes lose their status of דבר שבקדושה, and become permitted, the same way we read other Torah verses while praying alone. Such is the psak given in the Shulchan Aruch, ad loc.
In addition to not being a prayer as such, as mentioned in other answers, the shema is not a דבר שבקדושה (and we recite it while praying alone), hence according to Rashba, there's nothing lost by reciting the shema with trop.
Whether shema is ideally recited with trop or not seems to be a separate question, but in short, the Beit Yosef describes this (OC 61:25) as a relatively new practice, but that one should indeed use the trop. The Rama thinks it's unnecessary, and liable to be a distraction, but one may use the trop if one can without being distracted. Both these views are brought as psak in Shulchan Aruch, ad loc. See here for more.
1 The language is actually קריאה בעלמא, which could be taken to mean reading without the intention of בקשת רחמים, and not imply that trop needs to be used (à la קריאת התורה). However, it seems to me that the best way to make sure that קריאה בעלמא isn't a request for mercy is to read it explicitly in a different context, namely as if it were קריאת התורה.