Hold up. This is how Shove is supposed to work:
Using the Attack action, you can make a Special melee Attack to shove a creature, either to knock it prone or push it away from you.
Note: If you’re able to make multiple attacks with the Attack action, this Attack replaces ONE of them.
So it doesn't matter how many Actions you get, you can only Shove once per round.
When you want to grab a creature or wrestle with it, you can use the Attack action to make a Special melee Attack, a grapple. If you’re able to make multiple attacks with the Attack action, this Attack replaces one of them.
So, again, only once per round and it is an Attack Action, not a Bonus (but Larian is already doing that, so at least that's good).
You are incorrect in both cases. Shove and Grapple replace ONE of your attacks, and they are not capital-A "Attack Actions." They are attack-equivalent actions, which is a separate thing.
When you take the Attack Action, you are allowed to make X attacks depending on your class abilities. E.g., a level 5 fighter can take 2 attacks with a single Attack Action due to their Extra Attack ability. Each of these lower-case-a attacks can be replaced with a shove or grapple attempt. So a level 5 fighter can make 2 attacks, one attack and one shove/grapple, or two shove/grapples.
If this fighter has an additional capital-A Action (Action Surge), they can again take the Attack Action, which will give them another 2 attacks in this case for a total of 4. A level 4 fighter, however, only gets one attack for every Attack Action, so if they had 2 Actions they could only make 2 attacks/shoves/grapples.
tl;dr: According to 5e RAW, you can make as many shove/grapple attempts as the # of attacks you have. It is not limited to 1x per round.
