It's not about the rounds
From the combat rules, on reactions:
When you take a reaction, you can't take another one until the start of your next turn.
This means that there are situations in which one could indeed have two reactions in the same round. Using your example, if the player took an opportunity attack before his turn came up, then at the start of his turn he can now do a reaction again, even if it is still the same round. However, he can't take another reaction until his turn comes up again (next round).
To illustrate how this works, here is a round by round breakdown. In this example the player is called Bob, fighting against enemies E1, E2, E3:
- Round 1
- E1's turn (Bob's reaction is available)
- E2's turn (Bob's reaction is available)
- Bob's turn (Bob's reaction is available)
- E3's turn (Bob's reaction is available)
- Round 2
- E1's turn, Bob uses his reaction for an Opportunity attack
- E2's turn (Bob can't take reactions at this point)
- Bob's turn, he can now do a reaction again, including during his turn
- E3's turn, Bob reacts using his homebrew item
- Round 3
- E1's turn, (Bob can't take reactions at this point)
- E2's turn, (Bob can't take reactions at this point)
- Bob's turn, he can now do a reaction again, including during his turn
- E3's turn (Bob's reaction is available)
(and so on...)
The main point is that rounds don't really matter for the purpose of reactions. Like the quote says, you can take 1 reaction, and you only get the ability to do so at the next start of your turn.