Hello Armin,
fighting since 2 days with same problematic.
I have WIn7 x64.
As H2 is a x86 application, installed jdk-8u251-windows-i586.exe
Started H2:
Tried first with h2-1.4.200 and psqlodbc_12_01_0000, 11_01_0000, 11_00_0000 and 08_02_0100 (this version has no test button in the ODBC administrator).
Like in the documentation stays, added a DSN (PostgreeSQL Unicode) into System-DSN ODBC Administrator 32-Bit and got always same massage like yours.
Only 09.03.0300 gave me success:

Haven't done any further testing, because it made me so tired :(
What about your problem, have you or anybody any solutions?
Eduard