Drinking from a baby bottle 

Fill up three or more baby bottles of milk or juice. Ask for three or more volunteers and hand them the baby bottles. On the word "go!" they have to start drinking. The one who drinks it all first is the winner. 

Baby socks matching game 

You need as many pairs of baby socks as you can get (at least 10) for this cute baby shower game. Place the baby socks (separated) in a pile at the center of the room. Try getting socks in solid colors, patterned, and in different sizes). Each guest gets 10 seconds to match them. The guest with most pairs of matching socks wins a prize.

Baby animal game

Compile a list of proper names of baby animals, like fawn, gosling, kid, etc... Use a dictionary or a kids book; they usually have the oddest ones. And the odder the better! 
Make a list of 15 to 20 names, and the guests have to name who the mom is, like goose, deer, goat, etc... Whoever has the most correct names wins!

Guess the name of the baby 

Ask everyone to bring a baby picture of themselves to the baby shower. You can write it on the invitations. Upon arrival, the hostess takes the baby picture and arranges it onto a poster board. Assign a number under each picture. Each guest gets a piece of paper and is told to match the guest's name with the corresponding baby picture. The guest with the most correct matches wins! 
Don't forget to include a baby shower pinata in your list of baby shower party games.