So for a Fatshark, do I need to have both FPV transmitter and receiver antennas to be of the same brand and same specs? What about for a DJI goggle?
No, specific brand and specs don't matter for interoperability (aside from making sure that both antennae are built to transmit/receive the same frequency). Specific specifications about an antenna can help you figure out which will work best for your use case, like the antenna's gain, but mixing brands and types is fine.
However, there are certain mixes of antenna polarizations that will work better than others. You can learn more about this in Can circularly and linearly polarized antennae be used together for FPV? and about what antenna polarization is in What is circular polarization in antennae?.