That looks like the contract is the recipient of both types of tokens. The contract might have functionality to also transfer the tokens out of it but it might not. If it doesn't have such functionality the tokens simply stay there and can never be transferred out of it.
You can't implement both of the standards in one contract as they have overlapping function names - the function signatures are somewhat similar (maybe even the same, didn't compare too closely) but the actual implementation can only be for one token type.
Regarding token recipients
Any address can be the beneficiary, receiving ERC-20 and ERC-721 tokens (and also ERC-1155, ERC-777, and any other tokens).
Regarding token contracts (the thing that emits the events)
However, no, it is not possible for a token contract to both be an ERC-721 and an ERC-20 contract at the same time because those two specifications have overlapping and incompatible requirements.*
*I'm putting an asterisk here because ERC-20 is not well-specified. If you have a dog, then that dog is a compliant ERC-20 token. That is because ERC-20 does not say anything about what is required, and therefore your dog is meeting the (nonexistent) requirements. Most "ERC-20" contracts do not actually follow ERC-20.
*Also, if you were to imagine that ERC-20 was written in prescriptive language and it was actually a specification (i.e. hand-waving over my existentialism note above) then there is a single situation where an ERC-20 could could also be an ERC-721 contract. This is a degenerate case. Like in geometry class where your teacher that a point is actually a special kind of a circle. And that is if the ERC-20 and ERC-721 contracts both only have one token and its ID is 1.
Standard can be "retro compatible". However, I do not think it is the case for ERC-20 and ERC-721.
Indeed, they are related to different types of token, fongible and non fongible. So you cannot manage both at once in this case as function names are the same, but behavior is different.
However, if you are interested in both fongible and non fongible tokens managed in a single contract or by a single standard, I believe ERC-1155 may be what you are looking for (but not yet validated looks like).