bitcoin.org has a wizard that helps you chose your wallet:
https://bitcoin.org/en/choose-your-walletIt seems like there is some confusion about the basic concepts... I'll try to give a very simplified explanation about what is happening between seed/mpk/pk/address/derivation path/... mind you, this explanation is simplified.
Most recent wallets are either custodial (meaning: don't use those) or non-custodial. A custodial wallet usually doesn't let you access your private key(s), you're not in controll of your funds, so i'd advice against using those.
A non custodial wallet is usually a HD wallet nowadays.
Usually, they generate a seed phrase you can write down on a piece of paper in order to restore your wallet afterwards.
The seed phrase goes trough an algorithm to create a master private key.
Following a derivation path, private keys get derived from this master private key.
Usually, a public key gets derived from the private key, this public key gets hashed and the hash is actually the address.
Over the years, we moved from "default" wallet type. In the very beginning, we used pay to public key wallets. Then we switched to pay to public key hash "legacy" wallets, then to pay to Pay-to-Witness-Public-Key-Hash segwit wallets, and in the future even pay to taproot.
The thing is: the last 3 types of wallet (legacy, segwit and taproot) can be derived from the same master public key. That's why the same seed phrase shows wallets whose addresses begin with 1 (legacy), bc1q (segwit) and maybe even bc1p (taproot)
so, now, let's dig in to your OP:
For example, a wallet like Exodus allows you to save private keys of your wallets, but it also allows for saving seeds.
Let's say I save the private seed of my Exodus wallet, how can I be sure that in 20 years from now, Exodus wallet will still be a working product, that I can download from a new computer perhaps and import my seed into? This never made sense to me, I'm depending on the creator of the software it seems to me.
I'm not an exodus user, but if you read the above explanation, you could note that if you write down the seed phrase and use a wallet that supports the same "type" of seed (for example, a bip39 seed), the "other" wallet should handle the seed phrase in the same way, so it would generate the same master private key, and if you use the same derivation path, it should create the same private keys and addresses.
The private key of my Bitcoin wallet is different ofcourse, but I do wonder, will this private key allow me to import the address into other wallet softwares? Or do I have to import it on Exodus?
If you have the actual derived private key, you should be able to import it in many different wallets... not all tough, they have to allow you to import single private keys, and some wallets only allow you to restore a wallet from seed
One thing that also slightly confused me was the fact that when I request my private key from the Exodus wallet, it shows me multiple private keys and public wallets. Two public wallets starting with 'bc1q', and one wallet starting with '157okQ'. Which one would I have to save? When I make a request to receive funds in the wallet it shows my deposit address as one of the bc1q addresses. Will this be the address of which the associated private key is practically my money?
i actually went into this topic on the top of this very post