Connection pooling surely has that type of behavior, but the providers I'm familiar with wouldn't typically keep those connections active to reuse for days, or even hours. In fact, ADO.NET will eliminate the connection from the pool in about 4 to 8 minutes or inactivity.
Yes, it sounds like you have clients not properly closing and disposing of connections, which could be leading to more severe problems than just connection placeholders. But at the same time, you should not only be verifying
login_time, but you also have to consider
last_request_start_time to get an idea of activity of the session (or the lack thereof).
Is this standard?
It depends. For that duration, no that's not typical and surely warrants the investigation of how clients are managing connections to the data source.
should those conenctions that are sleeping be closed?
You would first need to determine that they shouldn't be currently active, and then you would need to confirm with the business and end users whether or not you can clean that up. But that would most likely be a very temporary fix, as if this is programmatic connections (think client applications) then it will probably come back once applications and/or end users reconnect.
Take a look at this blog post I wrote about connection pooling. It is relatively thorough and should give you a good idea on distinguishing between connection pooling, as well as poorly managed connections.