Is the arab empire and the islamic empire the same?

NO and yes too, most Arabic countries are the original North African countries (Tunisia, Egypt, etc., later countries like Saudi Arabia , Oman etc. were countries made possible by Berbers and Bedouins. The Islamic empire were countries in History with beliefs in and accepted by the faith of God and belief in Mohammed and called such by Europeans who were Christian in faith.

Islam is a religion, Arab is an ethnicity. The two empires you are referring two are different.

No, when the Arabs conquered Persia about the year 900, the Persians quickly asserted themselves as the Muslim masters and formulated an Islamic Persian empire that stretched into modern India and central Asia.

who even cares, their empires were kept alive by christians.

No

There are Islamic empires which were not Arabic. Islam is a religion, Arab is an ethnicity.

In general, the term Islamic Empire will be used when someone wants to lump different dynasties into one basket, without someone pointing out "yes, but..."

no