Well, it's worth noting that the number of neurons doesn't scale linearly with the brain size; in particular for two dogs analyzed by Herculano-Houzel et al. (2017), the number of neurons in the cortex scaled the least:
Both dog individuals examined (a 7.45 kg mixed-breed and a 32 kg golden retriever) had larger brains than the cat (brain mass in dogs, 58.4 and 114.5 g, respectively; cat, 34.8 g), and also more brain neurons than the cat (dogs, 1.8 and 2.6 billion neurons, respectively; cat, 1.2 billion neurons). The same applies to the cerebral cortex of the dogs, at 46.2 g with 429 million neurons and 84.8 g with 623 million neurons, against 24.2 g with 250 million neurons in the cat. Strikingly, although the cerebral cortex of the golden retriever was almost twice as large as the cortex of the smaller dog, it only had 46% more neurons than the smaller dog cortex (as expected from the non-linear scaling of cortical mass with number of cortical neurons [of carnivora in general]);
And the authors of this paper also write that this unsurprising because cortex neurons are apparently the most expensive to scale up in numbers:
We have previously shown that the metabolic cost of the brain is proportional to its number of neurons, regardless of brain size, and that neurons in the cerebral cortex cost on average 10 times as much energy as neurons in the cerebellum (Herculano-Houzel, 2011). Thus, cerebral cortical neurons are expected to be both more vulnerable to caloric shortage than other brain neurons, and to also contribute more to decreasing total metabolic cost when their numbers are reduced than the loss of other neuronal populations would.
And seemingly being a bigger carnivore with somewhat more cortex neurons might actually make one a little bit smarter. The paper notes in support of that that:
Cognitive performance in carnivorans was recently addressed specifically by Benson-Amram et al. (2015). Across these species, even though brain size relative to body mass is a significant predictor of success in opening a puzzle box, species with larger absolute brain volumes also tended to be better than others at opening the puzzle box (Benson-Amram et al., 2015).
(This paper might have been somewhat sensitive to the method used for testing, like the scaling of the box. It actually found that [unsurprisingly to me] bears were champions at opening boxes... but Herculano-Houzel's paper actually found that bears don't have that many cortex neurons... "the brown bear has [...] 251 million [cortex] neurons, which is only about as many as the house cat, even though the brown bear cortex had a nearly 10-fold larger mass". So something doesn't quite click here.)
Anyway, your hypothesis that a cat and a lion (or actually a bear) have the exact same cognitive abilities might not be correct, when generalized. (Lions have as many cortex neurons as dogs, on average, 500 million, according to Herculano-Houzel.)
There is also a 2019 paper by Horschler et al. on dogs specifically that finds something similar based on more data on dogs:
Using citizen science data on more than 7000 purebred dogs from 74 breeds, and controlling for genetic relatedness between breeds, we identify strong relationships between estimated absolute brain weight and breed differences in cognition. Specifically, larger-brained breeds performed significantly better on measures of short-term memory and self-control. However, the relationships between estimated brain weight and other cognitive measures varied widely, supporting domain-specific accounts of cognitive evolution.
I haven't looked at their methods section yet.