'To' in this sentence translates as 'this'. You need to use an appropriate word to match the gender of the noun, e.g.
masculine: 'ten mężczyzna' - this man, 'ten pająk' - this spider
feminine: 'ta kobieta' - this woman, 'ta książka' - this book
neutral: 'to dziecko' - this child, 'to zwierzę' - this animal
So in your example: 'Pająk jest tygrysem' would mean 'A spider is a tiger', 'Ten pająk jest tygrysem' would mean 'This spider is a tiger' and 'Pająk jest tym tygrysem' would mean 'A spider is this tiger' ('tym' is an instrumental case form of 'ten').
You're serious? "cat" is not a translation of "tygrys".
Also, "Kot to tygrys" is correct grammatically, but not semantically. "Tygrys to kot" is true, because tigers are big cats. "Kot to tygrys" = "A cat is a tiger" is wrong. Not every cat is a tiger. Just a really small percentage of them.