Did Jesus die for all? Is “for many” the same meaning as “for all “ in Matthew 26:28?

The answer is that both "the many" and "the all" are true. The invitation is to all. Any person from any tribe, or tongue, or nation may be redeemed by the blood of the Lamb and there is no particular state of a person which renders them irredeemable. That is to say, no one who earnestly applies to Christ will be told, "Nope, you are beyond saving". Also, the blood of the crucified Lord need not be shed afresh for each repentant sinner but has been shed once for all so that whoever believes may have their sin expiated rather than hearing, "Nope, you weren't accounted for in the sacrifice".

For Christ has entered, not into holy places made with hands, which are copies of the true things, but into heaven itself, now to appear in the presence of God on our behalf. Nor was it to offer himself repeatedly, as the high priest enters the holy places every year with blood not his own, for then he would have had to suffer repeatedly since the foundation of the world. But as it is, he has appeared once for all at the end of the ages to put away sin by the sacrifice of himself. And just as it is appointed for man to die once, and after that comes judgment, so Christ, having been offered once to bear the sins of many, will appear a second time, not to deal with sin but to save those who are eagerly waiting for him. - Hebrews 9:24-28

Notice in the above passage how Jesus appeared once for all and, in doing so, he bore the sins of many. A limiting factor has been introduced between the available efficacy of the redeeming blood and the application of the same. The limiting factor is not in the nature of the sacrifice Himself.

Some have placed the limiting factor within the will of God and say that the blood is efficacious for all but God chooses to apply it only to some. Others have placed the limiting factor within the will of man and say the blood is efficacious for all but only some will apply. Here is a simple chart comparing these two outlooks: http://www.crivoice.org/tulip.html This is beyond the scope of your question but is useful for pointing to the fact that a limitation has appeared between "all" and "many" and the fact that this limitation has spawned opposing theologies is strong proof of it's existence.

Enter by the narrow gate. For the gate is wide and the way is easy that leads to destruction, and those who enter by it are many. For the gate is narrow and the way is hard that leads to life, and those who find it are few. Matthew 7:13-14

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).