The sefer תולדות תנאים ואמורים here writes that every Rosh Yeshiva had one or more "Tannaim", even in the latter period of the Amoraim, whose job it was to be the resident expert in all the Baraissos (Tannaic teachings) by heart, so that whenever a Baraissa (which were generally learned by heart) was needed he was like a living book and would be able to immediately recite the required teaching. Thus the teachings of these "Tannaim" are referred to as the "Tanna D'vei Rebbi...".
The gemara in Shabbos 26a-b is a disagreement between Abayei and Rava with regards to the quoted teaching of R. Shimon ben Elazar and the teaching of the Tanna D'vei R. Yishmael:
Abayei holds that that they are in complete agreement, and that only a piece of cloth made from flax (or wool) is susceptible to tumah (as long as it is at least 3x3 fingerbreadths).
But Rava holds that they agree about a piece of cloth which is 3x3 fingerbreadths, but disagree about a piece of cloth which is 3x3 handbreadths or more. The Tanna D'vei R. Yishmael holds that even then only if it is made from flax or wool is it susceptible to tumah, whereas R. Shimon ben Elazar holds that in that case even if the piece of cloth is made from other materials, e.g. cotton, it is susceptible to tumah.
This gemara is explained in many online shiurim, for example here from 10:45 minutes onward.