I would think that Egypt had more in common with the Palestinian people than Israel, since both Egypt and Palestinians are mostly Sunni Muslims. Why did Egypt start the blockade of the Gaza Strip, and why are they upholding it?
TL;DR: Hamas (which rules Gaza) is an offshoot of Muslim Brotherhood, which is an Islamist group hostile to Egyptian government.
E.g., in an opinion of a local expert (Emad Gad, an analyst at the Egypt government-funded Al Ahram Center for Political and Strategic Studies) was quoted as:
Hamas is part of another coalition in the region – the Iran, Syria, Hezbollah coalition,... Hamas is a branch of the Muslim Brotherhood. And Hamas is trying to minimize the Egyptian role in the Palestinian cause.
Another factor is specifically security (but also stems from Hamas/Moslem Brotherhood connection). MEPC quotes Saudi sources as:
Some observers, including the editorial staff of the Saudi Gazette, have expressed sympathy for Egypt’s position, stressing Cairo’s need to defend its territory from Islamist militants: “Egypt's curbs on movement through its crossing with the Gaza Strip is a security decision that had to be taken even though it has cut off imports of medicine and aid to the impoverished coastal enclave....Egypt had no other recourse but to seal the tunnels although it is not a decision without physical and political risks. The ban and its broader war is an open invitation for extremist groups in Gaza and Sinai to continue striking at Egyptian targets....But the crossing had to be closed because Hamas had been inviting all sorts of militant and Jihadist groups and training them in Sinai, kidnapping and killing Egyptian soldiers and smuggling the killers into the Gaza Strip via tunnels and hiding Muslim Brotherhood leaders in Gaza.”
The answer about the Muslim Brotherhood is absolutely correct. But there is more.
Relations between Egypt and Israel are excellent. Al-Sisi has personally met with Netanyahu. And Israel and Egypt have been jointly fighting against Islamists in Sinai, who have been responsible for a recent massacre at a Sufi mosque and also an airplane bombing that killed a lot of Russian tourists.
Additionally, Hamas is heavily supported and funded by Iran. Relations between Egypt and Iran are not good. Egypt has good relations with Saudi Arabia, which absolutely hates Iran and appears to want to work with Israel against the common threat.
The upshot is that, while it is true that Israel and most of the Palestinians are in conflict, it is not true that Israel is in conflict with all, or maybe even most, Sunni Muslims in the Middle East. Al-Sisi in particular appears to agree with the Israelis that Iran, the Palestinians, and ISIS are causing trouble, and this is undesirable.