"We have more animals than them."
Translation:हमारे पास उनसे ज़्यादा जानवर हैं।
ज़्यादा, when not used as a comparative adjective, is closer in meaning to many/much.
वहाँ ज़्यादा लोग नहीं है। - There aren't many people there.
मेरे पास ज़्यादा समय नहीं है। - I don't have much time.
और, on the other hand, implies "more than what there is" if it precedes a noun or a verb, and intensifies the adjective if it precedes one.
मुझे और पैसा चाहिये। - I want more money.
और खाओ! - Eat more!
थोड़ा और पानी दो। - Give (me) some more water.
और कुछ (चाहिये)? - (Do you want) anything else (~more)?
इसे और ज़्यादा गरम करो। - Make it even hotter. (Compare with इसे और गरम करो। - Make it hotter.)
क्या यह और कम हो सकता है? - Can this be any less?
But we learned before that 'paas' is only needed for small objects. So couldn't you leave it off then? You say the sentence would then mean Ours are more animals than them, but since Hindi doesn't have a verb for 'to have' couldn't you translate all these sentences that way literally? We have three daughters/To us are three daughters.