I could resolve the problem:
The schematics displayed here Grove Shield for Arduino Nano | Seeed Studio Wiki
shows a connection between the RST - RST pins.
If you remove that connection, the Grove Nano shield works fine with RP2040 Connect board.
I removed the connection by drilling a small hole directly opposite the middle jumper of the 3.3/5V switch. Be careful not to cut the connections of the 3.3/5V switch on the other side of the board:
