You can get the powerful regular expression operators ~ and ~* for case insensitive:
WHERE Field ~* '(cat|dog|animal|pet)'
SIMILAR TO is case sensitive, so you'd have to do:
WHERE Field SIMILAR TO '%([Cc][Aa][Tt]|[Dd][Oo][Gg]|[Aa][Nn][Ii][Mm][Aa][Ll]|[Pp][Ee][Tt])%'
ANY ARRAY will work too, but the perfomance is worse:
WHERE Field ILIKE ANY (array['%cat%','%dog%','%animal%','%pet%'])
With some dummy data with 1000000 rows with a BTREE INDEX in Field, I get these results:
╔═════════════╦═════════════╗
║  Operator   ║ Time (secs) ║
╠═════════════╬═════════════╣
║ ~*          ║ 1.5         ║
║ SIMILAR TO  ║ 1.4         ║
║ ANY ARRAY   ║ 4.0         ║
║ OR OR OR... ║ 4.0         ║
╚═════════════╩═════════════╝