Yes, fire did burn differently because of excess oxygen fuel. | ELI5

Yes, fire did burn differently because of excess oxygen fuel. During the carboniferous period, 300-350 million years ago, oxygen levels reached as high as 35%, compared to today's ~20%. This allowed fires to start in places which we wouldn't normally expect (places of high humidity, during rain storms). Today we find fossilized charcoal in coal beds that arose from peat bogs, which grants credence to the idea that fires did burn much more easily during this period. You might ask "how did plants survive the constant danger of fire?" Adaptation thanks to random evolution. For example, plants from this period expressed deeper roots/tubers than their modern day counterparts, and the leaves of trees were higher which could help avoid catching fire from a stray brush fire.
The reason so much coal (a large part of which is oftentimes fossilized charcoal) remains from this period is due to two factors, as other posters have mentioned. First The structural material of plants, lignin, doesn't burn rapidly, (paper is often stripped free of lignin which is why it burns so easily) rather it tends to smolder (i.e. logs in a fire), so fires would leave large amounts of charcoal rather than their usual main byproducts: carbon dioxide,carbon monoxide (at lower burning efficiencies) and water. Secondly, lignin is very hard to digest, even fungus/bacteria today have a hard time digesting it. 300 million years ago, the decay rate was practically zero. In summary, pretty much all of todays coal comes from plants which, millions of years ago, died and were buried without ever going through the process of decay.
What's even cooler though is that during this time gigantic insects roamed the earth https://en.wikipedia.org/wiki/Meganeura which is believed to be related to the faster metabolic rates that high oxygen content allows.