It depends on how you define Palestine.
If you define it as the territories under the de facto control of the Palestinian Authority, it only borders Israel.
If you consider only the territories of the West Bank, even those under de facto Israeli control, it borders both Israel and Jordan.
If you consider all of the Palestinian Territories, also including the Hamas-controlled Gaza Strip, it borders Israel, Jordan, and Egypt.
If you consider the former Mandate of Palestine that many Arabs use when they discuss Palestine replacing Israel, it would border Jordan, Egypt, Syria, and Lebanon (but no Israel since Israel would be gone).
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The Gaza Strip borders Israel and Egypt.
The West Bank borders Israel and Jordan.
In order of longest to shortest border as the countries currently exist de facto: Jordan, Palestine, Egypt, Syria, and Lebanon.
Israel/Palestine, Jordan, Lebanon, and Syria are the four modern countries that have territory from Ancient Palestine.
Palestine
The two states border each other.
There are over 50 Islamic countries, so Palestine is certainly not the only Islamic country, but if you are considering Israel and Palestine, yes, Palestine is the Islamic country while Israel is the Jewish country.