UPDATE: Christopher Anzalone of Views from the Occident sent me a message clarifying the issue with the logos stating:
The logo in the video is different from JaTwJ’s, though the logo that appeared in the thread where it was posted was JaTwJ’s (perhaps used as a generic marker). The video intro editing is similar to other Gazan groups, making my hunch that there is fluidity in the various groups and that they share a “media wing” (probably a handful of shabab) even stronger.
NOTE: This is the first time I have ever heard of the Islamic Emirate of Gaza. The logo used by this group appears to be similar to Jāma’at Tawḥīd wal-Jihād, a Gaza-based global jihadist group that sympathizes with the al-Qā’idah worldview. Therefore, they might have changed their name to sound more legitimate.