Could this also mean "The nuns like my cat"?
Yup, it could.
Could it not also be the sister singular? How do we know its not?
die Schwester = the sister
die Schwestern = the sisters
In addition to what christian said, I believe you can also tell by the way mögen is conjugated.
How do we know its "sister" or "sisters"? There won't be any change in other words if its plural or singular would it?
Not sure what you're asking. The following was already posted above:
Schwester - sister
Schwestern - sisters
So, there clearly is a discernable difference between singular and plural.
Also for singular it is "Die Schwester mag meine Katze"
Couldn't it also mean "The sisters like my cat (singular)"? Katze is feminine, so it would take "meine" wouldn't it?
"meine Katze" only means "my cat", not "my cats".
meine Katze = my cat
meine Katzen = my cats
Why not to translate it as "love"? Well, moegen = like, lieben = love, but is "love" that wrong in this case?
Yes, that would be an incorrect translation. "Mögen" definitely does not translate to "love."