Your question is really two questions, the first is theological, the second is whether "all" and "many" are semantically synonymous enough to be interchangeable.
The second question is addressed first, because, properly understood, the Scriptures themselves will answer the first question. The adjectives, "many and "all" are not grammatically synonymous in either English or Greek ("polloi", "pas"). While both are indefinite adjectives, and used to describe/distinguish/emphasize some property of a (contextually) defined set, they address different properties. "Many" addresses the relative number in the set (not "few"). On the other hand, "all", connotes inclusion, without exception (not "some"). For illustration, many cars are red, but not all cars are red. Some are blue; some are green; a few are pink, in which, the contextually defined set is cars, and the property is color. Also, note, once having been defined, the set's identification/description is usually assumed for subsequent adjectives. This results in "floating" adjectives (an adjective without an explicitly associated focal-object). "Floating" adjectives are common within the the received (Greek) text, especially within long, complex sentences, and especially when the set's description is long, cumbersome, complex or complicated. Accordingly, in order to facilitate readability, as well as, grammatical-acceptability, translators sometimes render "floating" indefinite-inclusive adjectives as indefinite, inclusive pronouns ("all" or "each" is rendered, "everyone/everybody/everything"). A second grammatically acceptable option is to insert a plausible, numerically-indefinite noun or pronoun, immediately after the "floating" indefinite adjective. Interpretation errors result when this inserted word is inconsistent with the contextually-defined focal-set. Moreover, proper interpretation is especially difficult when the inserted word is not identified as having been "inserted". Since God is not the author of confusion (1Cor 14:33), nor ever errs or contradicts himself, erroneous insertions or interpretations cause (apparent) contradictions.
Regarding your first question, "Did Jesus die for everybody?.
1) If "many" is never synonymous with "all", because "many" never implies without exception, the answer to your first question is implicitly negated within your (Mt 26:28) quotation, namely, Jesus died for many (some, not everyone). Other similarly relevant uses of "many" include Mt 26:28 (ransom for many); Is 53:11-12 (many accounted righteous...bore the sins of many), Rom 9:28 (to bear the sins of many).
FYI: The "many/some" are elsewhere identified: Mt1:21 (His people), and Jn10:11 (for the sheep)
2) The phrase,"died for everybody" can not be found in any English Bible, nor the Greek equivalent.
3) So then, if Jesus only died for "some", what about (seemingly) contradicting verses like 2Cor 5:15, 1Tim 2:6, Heb 2:9 (died for all/everyone/all people), (ransom for all men), (tasted death for everyone)? In all three verses, a "floating" indefinite Greek adjective is either translated or interpreted outside of its locally-defined context. In all three verses, the local, contextually defined-focus is exclusive (not "everybody", nor "all men" (as usually interpreted and/or inserted). Accordingly, these verses illustrate how a strong "proof-text" for limited atonement is converted into a strong "proof-text" for unlimited atonement: by translating or interpreting one "floating" adjective, outside/beyond its locally-defined focal-context).