According to the parallel version brought in Shemot Rabbah 40:4, the word "ישי" (Jesse) in Tanchuma is a scribal error and should be "יש" (there is):
"יֵשׁ שֶׁנִּקְרְאוּ לוֹ אַרְבָּעָה זֶה אֵלִיָּהוּ"
"There is one who had four names, this is Elijah."
This is also evident from the next paragraph (in both texts), which begins discussing the list with Elijah, then continues to Bezalel, who was the main focus of this midrashic exposition.
This was also the version that was before Rabbi Avraham Helin (Hebrew Wiki) in his commentary Yedei Moshe and Rabbi Chanoch Zundel in his commentary Etz Yosef.1
It's possible that the scribal error in some versions arose from the fact that at the end of the list Elijah seems to be mentioned once again as having four names, however this is an erroneous reading of the text. Here Elijah is mentioned again as a dibbur hamatchil, a starting quote before the next section of exegesis, which discusses the origins of Elijah. And then the midrash moves to discussing Elijah's four names.
1 Note that the linked printed edition which features the Etz Yosef has the same scribal error; the Etz Yosef himself used a different edition, and perhaps also saw manuscripts.