I got to dig a bit into the Cambridge Economic History of the Greco-Roman World for this answer, which was fun, and I can speak a bit to olive oil production then and now. I think there's two fairly simple pieces of information that answer your questions.

There was definitely a LOT of olive oil production in Roman antiquity. Amphorae and olive processing equipment (i.e. olive presses) are all over the archeological record, and there's robust evidence of wide-spread production and export of olive oil at commercial scale - particularly from Spain and north Africa.

The first key element here is that olives were a really practical crop in the Mediterranean throughout the Roman period. Olive trees are relatively easy to care for. They do well in poor, dry soils and are tolerant to drought, but intolerant to frost. During the Roman Warm Period (~250BCE to 400AD), the European Mediterranean was drying and warming - trees like lindens and elms were pushed north, but the viable range for olives (and grapes) also expanded greatly. Options for alternative fats seem somewhat limited - Romans generally didn't keep cattle for dairy (though they did have dairy from sheep and goats); they had rapeseed and did produce rapeseed oil, but unlike rape, olives could be intercropped with wheat and grapes. On subsistence farms, this kind of intercropping (called cultura mixta) offers more variety in nutrition, and - according to Dennis Kehoe in the Cambridge History - on the larger landed estates, agricultural diversity is a matter of economic security for elite landholders. Certainly, rape would be subject to similar drought risks as wheat, which would differ from olives and grapes. Finally, olive trees take 10-15 years to produce a crop, and the infrastructure for larger commercial production is a significant investment, but once those investments are made an olive orchard can produce for generations - thus olive oil production in the roman era benefited greatly from the peace and security of the empire.

The second element - and this is ahistoric and a little banal, but very relevant to your question - is that 98% of the olive oil in the world still comes from the Mediterranean, but there are way more people consuming olive oil! According to figures from various oil olive industry publications I looked up, the average American consumes about a liter of olive oil a year (actually, ~50% of Americans consume about 2 liters of olive oil per year, and the rest don't), but all this oil, along with the olive oil everyone else in the world is consuming, is still predominantly coming from 7 olive oil producing nations around the Mediterranean, with Spain by itself producing 43% of the world's olive oil while only consuming 19%. Meanwhile, modern southern Europeans are consuming an average of 15-25 liters of olive oil a year. One paper I found guesses that roman consumption likely rivaled modern consumption in the region, while another guesses that any estimate is probably low (though neither justified those guesses as far as I can tell, so it's just speculation) - but if we do assume 60 million residents of the Roman empire were consuming over 1 billion liters of olive oil a year, that's only about a third of what the same region currently produces. Given what we do know about the prevalence of olive oil cultivation in the Roman world, it seems to me to be as simple as "they really just did produce a whole lot of olive oil and it went into everything".