No.  Each wallet implements its own method of storing private keys, seeds, and transactions.

The private keys are generally (though not ALWAYS) the same, so you can usually export the private keys from one wallet and import them into another when necessary.

Also, MANY wallets that use mnemonic seed words are adopting the same system, so those seed words are often cross compatible to other wallets that use that system.  Furthremore, wallets such as Electrum are implementing the ability to recognize many formats from other wallets.

In other words, you will often find inter-compatibilites between various wallets, but unless you've looked into it and verified that an inter-compatibility exists between two wallets you should NOT assume that it does.