We have a server which we connect through and do remote tasks. However, around 4-5 people login with the same username and password. Can this situation cause anything weird?
Define weird ..
You will not be able to use history, or other last-used settings, in a meaningful manner as anyone could have used/set these last. You will also be unable to assign responsibility, if anything goes wrong.
I don't recommend this practice, unless you don't care what happens to this server (and its content).
The SSH sessions themselves will be fine.
I have maintained more than one session, when monitoring command-line transfers or configuration, with no issues.