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If you stay with the same lender they will typically add a second account or portion of mortgage calling it a Further advance, homeowner loan or something to that effect.
When remortgaging to a new lender, some just give you one single mortgage to replace the previous one and give you the surplus, some separate the purposes, giving you one loan for replacing the current, and the additional money on a separate account (at the same/similar rate).
No real difference other than sometimes more complicated mortgage statements.
I had a client with Nationwide who got 14 separate Nationwide mortgage statements a year due to 13 different borrowing requests for historic house moves, home improvements, and so on.
This sounds a different process to mine. I just got one new mortgage that covered the old mortgage and had some extra.
This was with Tesco Bank (who no longer exist).