You say that there was no mention of there being only two cats, yet you conclude that he is the biggest cat. There is a problem with that, because you have not considered the posibility of the cat being intermediately large (larger than the smallest cat, but smaller than the biggest cat). The way it is written in German (größere insted of größte) prevents you from translating that meaning to biggest/largest in English. You can probably get away with it in other languages, but the adjective rules in English and German have common ground here and the translation would lose its meaning if you assume it is the biggest.
But again, I'm being prescriptivist and language changes over time so I don't know...
But in english the comparative of superiority need another subject to be compared,; while when you use the superlative you don`t need another subject, because the one you are refering is over all. So is not correct to say : " My cat is the bigger or the larger" , but "It is the largest" or "the biggest", or "it is bigger than...
Predicate vs attributive adjective, maybe?
Die Katze ist grösser. Es ist die grössere Katze.
The form of the adjective is fixed when it's on the other side of a verb, but it is declined when it's next to the noun. Doesn't matter whether it's a comparative like "größer" or not; it holds for any adjective.
The specific ending depends on not only gender, number, and case, but also "strength" or definiteness. That last category means that "big cat", "a big cat", and "the big cat" are different situations that potentially have different endings on the word for "big".
If there is an article an the word is pl., it is - en. Die großen Menschen Der großen Menschen Den großen Menschen Den großen Menschen If it is an nominativ in singular it is -e. Der große Hund Die große Katze Das große Schwein. Gen - en: Des großen Hundes Der großen Katze Des großen Schweines Dat - en: Dem großen Hund Der großen großen Katze Dem großen Schwein. Akk: Den großen Hund Die große Katze Das große Schwein
I don't understand what the "the" has to do with picking "larger" or "largest".
My cat is the larger cat. My cat is the largest cat.
Both of the above are correct English sentences. In formal written English, the first one is required when there are only two cats, but the second sentence is more common idiomatically.
You could avoid the redundancy of repeating "cat" a couple ways:
My cat is the larger one. Mine is the larger cat.
Or even just:
My cat is the larger.
That's still correct, but it ratchets up the formality level even more than just using "larger" in the first place.