Currently something like MSN Messenger Pets doesn't exist as said in this thread (it is also almost impossible to make something like that as suggested here)...
However!
Currently there is something called
Eccky aka
"create your own virtual baby on msn". This was launched on the 26th of October.
With Eccky, you create your own "MSN Baby" between you and a contact. Parents can chat (also via SMS) with their Eccky, play games, feed, wash, shop, and everything else that comes with raising a child in real life. Thus the whole idea behind it are the infamous Tamagotchi games...
Though, since this is kind of a game, it has also an end. This will be after 6 days when Eccky will celebrate its 18th birthday. After having gone through three life stages, baby, toddler and teen, Eccky will leave its parents home...
All interaction is done either through the MSN Messenger chat interface, the MSN Messenger game window or by using a mobile phone.
The Eccky Game is currently (as far as I know) only available in Holland. It is also not free, one of the parents must pay €1.5 thru SMS before Eccky is born. But, from that point on you both will recieve child benefit (or children's allowance), this means you will recieve money back to buy cloths and other stuff you can use for your baby...
Thus, for all who speak dutch, go to http://nl.eccky.com/eccky_nl/home
and check it out 
For an explanation of what you can do and all see:
Official Dutch explanation:
http://nl.eccky.com/eccky_nl/Hoe+werkt+Eccky/.
Short English explanation:
http://www.eccky.com/weblog/2004/08/wtf-is-eccky.html.
Thus, also see
http://www.eccky.com/weblog/ (includes some English texts also)...
EDIT: fixed mybb tag error
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