Two breakers sharing same neutral?
Updated handle ties for double pole breakers were added in 1981 I was confusing that with multiwire branch circuits because the local jurisdiction added mwbc ties the same year but the NEC required handle ties in 99 or 02 if they landed in the same box or yoke and later in 08 all mwbc’s required handle ties for the reasons listed below.
I believe you have tandem breakers looking at the double pole compared to the other breakers, if they are tandem your setup would be ok. 4 breakers on 2 poles the left 2 are on leg1 the right 2 on leg2  they make ties to connect the outside 2 and clips or brads for the inside 2. Back when your home was built they were not required and you do not have to update them. If it was my home I would swap the wires so the mwbc was on the center 2 and cut a 6 penny nail so when it is put in it doesn’t extend below the bottom handle this will make it safer so anyone working on either of those 2 circuits doesn’t get a Wake up call from mr zap. But I would verify they are tandems so you know you are on separate legs in any case (measuring the 2 Hotta should show 240v that’s good, if 0 it will need to be moved.)
The 2 breakers being tied together on a 240v service means they are on separate legs. This is needed because if you want to work on circuit one and they were not handle tied you can get knocked on your butt or worse because the neutral is the return path and is carrying the voltage back to the source so although not required on your home it is a good idea
Having a common neutral is legal and common to do this it is called a multiwire branch circuit.
Both share the same neutral. They can do this because with 240v single phase leg 1 is at the maximum positive voltage while leg 2 is at a negative max they actually cancel each other out if exactly the same size the imbalance is returned via the neutral.
With all the AFCI and GFCI requirements multiwire branch circuits are not used as often because of false trips, they they are /were used is you get a second circuit for 1 extra wire this can save a lot of $ especially with a large home or on a longer run to the panel.