Could you elaborate how another wallet provider does recognize what wallet has to be restored when typing in the 12-mnmemoric seed phrase?
Bitcoin users can use different wallet softwares and they can import mnemonic seed created by one wallet software to another wallet software if two wallet softwares are compatible with each other and support similar BIP used in wallet creation.
If a second wallet software does not support BIP used in a first (initial) wallet software you used to create your wallet originally, you can not import that wallet to a second wallet software.
You can import Private keys or Mnemonic Seeds. Examples.
https://bitcoinelectrum.com/importing-your-private-keys-into-electrum/https://bitcoinelectrum.com/restoring-your-standard-wallet-from-seed/