A general preface: each edition of D&D is its own game, and rules should be evaluated as such. Just because things happened in old rule sets does *not* mean that it will in 5e. Though it can be helpful to look at old editions for inspiration, you should be careful of the rules environs of the edition you are trying to modify when making house rules.

In 5e, first, Shield Bash is already a named mechanic: it's the additional power associated with Shield Master that allows you to attempt to push someone as a bonus action when you make an attack. Also, you can do this same thing as a Full Action (or a piece of an attack action) using the contest rules and the "Shove" improvised action (Players Basic 74). That said, that's not what you're asking about here. What you want to know is how to do damage with your shield.

If you want to hit someone with your shield, it is treated as an improvised weapon attack.

The question then becomes *does it resemble any existing weapon we have game statistics for?* as per the "Improvised Weapons" section on pg 47 of Player's Basic. Looking at the list, I don't see an obvious resemblance (though feel free to make your own observations/additions). Thus it acts as a 1d4 weapon and proficiency is only granted if you have proficiency in improvised weapons. 

There is also no provision for removing the defense bonus. In fact Shield Master *improves* the shield with attacks *without* removing the defense bonus so there seems no need to import that rule from a prior edition. (Additionally, it seems to be punitive beyond the spirit of 5e's rules.)

So, ultimately, shield bashing is a flavorful power that is probably not a good mechanical choice. If you want to do it, talk to your DM, but without house rules, it's not a great use of an action. If you want the mechanics, and don't mind not doing damage on the push, Shield Master's push is probably the best you're going to do.