Before digging into the details; I just want to make it very clear that enterprise databases like oracle are designed to do hundreds of things simultaneously... so doing just one thing from your app at a time is definitely not necessary. Similarly, micro-services are used to handle thousands of request a second in parallel in many places.
Do I have to be careful when 2 instances of A try to access the oracle db at the same time?
This completely depends on the operations you are running against the database and how you are executing them. If you use database transactions, even complex multi-statement workflows can be isolated/safe when done in parallel (its very normal actually).
But... if I sent two parallel requests against your micro-service(s) and it resulted in two threads (regardless of which app) trying to create the same object in the same table, then one of them would beat the other one and the other would fail (assuming your database has proper constraints).
This could even happen with one instance of your service A if you allow it to have multiple database connections (which is likely if you're using a library).
So... if you have multiple instances of the micro-service talking to the DB, you probably can't easily control making sure they don't try to do the same thing at the same time, so you must have defensive coding to detect and handle when duplicate data is sent to the DB, etc. Also, you must be aware that the DB state may change due to an app outside of this one and reload data if you cache anything periodically.
Also, what would happen if I have only 1 instance of A and 2 instances of B which try to send serialized objects to A at the same time?
Generally this would be normal/expected. The idea is you spawn up more copies of your micro-service as they are required due to load. Normally micro-services handle multiple requests at a time whether it is by REST, sockets, etc.
So, the requests wouldn't wait... unless you purposely coded your one instance of A to have a single database connection and to handle requests in a queue. But again, that's down to your personal implementation details.
