Is it bid'ah to say "Eid Mubarak"?

What bid'ah actually means

Bid'ah means inventing something in the religion: a new act of worship, a new ritual, or a new belief presented as part of Islam. The Prophet ﷺ warned that whoever introduces into our matter something that is not part of it, it is rejected (Sahih al-Bukhari 2697). Scholars apply this to worship. Everyday customs such as clothing, food and the words people use to congratulate each other are a different category, and the basic rule for customs is that they are allowed unless something forbids them.

Why "Eid Mubarak" is a custom

• Nobody says it as a required ritual or believes it has a special reward on its own.
• It is simply a good wish: "a blessed festival."
• The Companions greeted each other on Eid with "May Allah accept from us and from you," which shows that congratulating people on Eid has a basis in the early community.
• Scholars such as Ibn Taymiyyah said Eid congratulations were practised by some Companions and allowed by Imam Ahmad and others, and that the exact wording is flexible.

Imam Ahmad himself reportedly said he would not start the greeting, but would reply to anyone who greeted him. That shows its relaxed status: allowed, not required.

Answering a greeting is good manners

When someone wishes you well, Islam teaches you to answer kindly, even better than they did.

What about people who say it's wrong?

A few voices online call any greeting not found word for word in the hadith a bid'ah. This is a minority view, and it isn't how the major scholars of the four schools have treated festive greetings. If someone near you prefers "Taqabbal Allahu minna wa minkum," you can use that with them out of courtesy, but there's no need to feel guilty about "Eid Mubarak." The same discussion comes up with Friday greetings; see Is saying "Jummah Mubarak" a bidah? For the meaning of the phrase itself, see What does "Eid Mubarak" mean?