Would a square piece of copper have a similar maximum current rating as a copper wire?
A quick and dirty, approximate way to approach this would by by using an ampacity table rather than a fusing current table. As you observed, your conductor has approximately the same cross-sectional area as 8 AWG wire. Ampacity tables for 8 AWG wire show current limits ranging from 40-50 Amps or so.
Ampacity tables are designed with the goal of allowing you to use wire and be confident that it will not exceed the insulation temperature rating. So they generally assume that the wire is in conduit with other wire. Also, they are for long wires. So they do not strictly apply to the situation described in this question. But it still gives a rough idea.
If it is acceptable in your application for the copper cube to become hot, then it is likely that you could safely exceed 50 Amps. The fusing limit given in the chart in the question is DEFINITELY too much current and could risk actually melting the copper.
If your use case involves intermittent high current, the situation may not be too bad. It takes some time for a piece of copper to heat up. So it can probably withstand the fusing current for 10 ms or more without melting. But it may require a long time to cool off again before the next pulse.
In situations like this, experimentation is what I would recommend to gain confidence. Take reasonable precautions during your experiments to make sure you don't get hurt or burn down a building or something. Keep your distance, use protective gear or barriers, and make sure you can de-energize the device remotely (like with a switch or breaker or whatever). Do not put the device between you and your exit path (in case you need to run away). Keep a fire extinguisher handy.