Does it matter which bank I choose and would it make sense to close down my OCBC Frank if I use DBS Vickers or vice versa? I'm getting started on investing.

You should look at high interest savings account once you start working. If you can hit their requirements, they offer better interest.

https://seedly.sg/tools/savings-account-calculator

Dont limit yourself to DBS and OCBC just because you have accounts with them. Check the various fees that they charged for investing with them. Fees will eat up your returns. Why need to close the account if you use another broker? Just close it if you dont want to use that account anymore.