I am developing a software which will connect to the same database as another application. Is there a term for that?

Also any tips on what sort of methods I can use?

Another question, if I was to use two separate databases.

What could I use to push the data from one database to another(i.e syncronise)?

Database Replication is one term for synching databases - particularly when talking about SQL Server. Not sure about the other term - could be multi tenant?

I am developing a software which will connect to the same database as another application. Is there a term for that?

I cant think of a techincal term, but in any case, this is that uncommon. Databases are generally built to allow multiple connections from various sources/clients.