Could you elaborate how another wallet provider does recognize what wallet has to be restored when typing in the 12-mnmemoric seed phrase?
You should know that your wallet software acts as a tool to generate your keys and helps you interact with the blockchain. Your seed phrase holds the access to your digital assets allowing your to recover them through any compatible wallet.
Upon restoration using a 12 word seed in a wallet like Electrum, the software then uss Electrum standards and derivation methods like BIP44 or its own to define your key pairs and associated addresses. It then scans the blockchain for these addresses, and the balance related to it.