Thank you for explaining how Shield Mastery works.
In what way is shoving with your shield not a bash with it?
Because a shield bash involves an attack roll and potential damage. Shield Mastery allows you, when you use the Attack action, to use a bonus action to Shove using your shield. Shove is, technically, a thing that can be done using the Attack action that is not an attack - you forego the attack roll and possible damage in exchange for contesting your Strength(Athletics) against the target's Strength(Athletics) or Dexterity(Acrobatics) and, if you win, you can knock the target prone or push them 5'.
Suddenly fighters with shields prefer the Two weapon style in order to add their ability modifier to damage with the shield.
Certainly some will, but they will need a total of two feats to do it:
Dual Wielder (to allow them to use Two Weapon Fighting with weapons that are not light - a shield is not light)
Tavern Brawler (to allow them proficiency with the shield as an improvised weapon. Otherwise they would be attacking with disadvantage every time and not adding their proficiency bonus)
Shield masters get to pick whether they deal damage or shove the opponent as their bonus action.
No, Shield Masters who are also have the Tavern Brawler
and Dual Wielder feats get that choice.
I can choose a light weapon dealing 1d6, or a shield dealing 1d4 as well as AC, etc, if I take a feat, in my off hand.
As above, no. You need two feats to make it work.
I think I'll pass that house rule, but to each her own.
It's not a house rule, it's the RAW. Now, you can house rule that this is
not possible in your game and that's fine. Personally, I think two feats and a number of class options sunk into making a mechanic viable is enough of a price to balance it. It's reasonable enough to be totally viable to build a character towards this path, expensive enough to be rare and, while it's a nice effect, it's not so powerful that every character is going to want to do it. It makes an interesting flavourful choice to build a character around, while being mechanically viable.
At this point, I'm just going to go on to what's become my standard refrain of taking this as an example of why 5e is such a great, flexible system. I'll spare you the details
