Based on you example I think you mean to say:

ううん、違う犬がいる。 No, (they have) a different dog.

持つ does mean 'to have' as well as 'to hold'. However in terms of living things, ownership is better expressed with いる as in 'to exist'.

There are actually many ways to express difference/different/being different and depends on what you are trying to express.

Off the top of my head, two widely used ways to express 'different' would with 違う and in addition 変わる.

変わる　ー　be different (from)

If someone is confirming the type of dog, 違う would be appropriate.

彼らプードルがいる？ Do they have a poodle? ううん、違う犬がいる No, they have a different dog. (maybe a shiba-inu)

And for example, someone is asking what type of dog, 変わる can be used.

彼ら面白い犬がいる？ Do they have a fun/interesting dog? ううん、変わった犬がいる No, they have a different/weird dog.

Based on you example I think you mean to say:

ううん、違う犬がいる。 No, (they have) a different dog.

持つ does mean 'to have' as well as 'to hold'. However in terms of living things, ownership is better expressed with いる as in 'to exist'.

There are actually many ways to express difference/different/being different and depends on what you are trying to express.

Off the top of my head, two widely used ways to express 'different' would with 違う and in addition 変わる.

変わる　ー　be different (from)

If someone is confirming the type of dog, 違う would be appropriate.

彼らプードルがいる？ Do they have a poodle? 
ううん、違う犬がいる No, they have a different dog. (maybe a shiba-inu)

And for example, someone is asking what type of dog, 変わる can be used.

彼ら面白い犬がいる？ Do they have a fun/interesting dog? 
ううん、変わった犬がいる No, they have a different/weird dog.