Lucy get Plue's key from a shop (at least in the anime) and seems to keep Plue around with her all the time. It is often seen hanging out with Happy, Natsu and Lucy. However, after
Lucy discovers that Loke is Leo the Lion, a Celestial Spirit
we learn that Celestial Spirits cannot stay in the human world too long. So why is it that Plue can be around all the time?
I'm going to take a stab at your question. You asked...
Why is Plue always around?
Initially Plue was only contracted to be around on Tuesdays and Thursdays as that was when he was available. In the Fairy Tail Wikia it says this...
...When Lucy first contracted Plue, he only agreed to be summoned on Tuesdays and Thursdays. Later, however, Plue is available anytime, often appearing at times other than his initial contracted days. Because he is not designed for combat, he serves as a cute pet for his master instead.
Plue cannot stay all the time, he uses his own magic to cross over from the Celestial World as displayed by Virgo and Loke. Also you could think of Plue as a pet in some way. He was not made for fighting and it is plausible to say that he can stay in Earth World for as long as a month or even a year, just like Loke has demonstrated in the past.
Plue is always around because he loves Lucy and being around Family. That's my answer. The Reason behind this is...
Plue enjoys sweets and shares a close friendship with Happy. Their good relationship could be accounted for the fact that they are always together during Team Natsu's missions, as well as their almost similar attitudes. Happy and Plue often tease Lucy and Loke