It may be the case you need to set some environment variables, perhaps in an ~/.Rprofile.
(My situation is rather different in that I am using ROracle instead of ODBC and that package won't load in RStudio. In my case the environment variables don't even seem to matter, but it might help in your situation?)
Based on your hint I checked the values of the three environment variables in CLI R (connection is working from there) and in rstudio (connection not working):
TNS_ADMIN is empty in every case
LD_LIBRARY_PATH is identical
ORACLE_HOME points to /opt/instantclient_19 in the working environment. It is empty in rstudio.
I thought I had found the problem and tried the following:
Sys.setenv(ORACLE_HOME = "/opt/instantclient_19")
Create a wrapper script for rstudio with export ORACLE_HOME = /opt/instantclient_19
Both attempts (and a lot of similiar messing around with environment variables) don't change anything. I still get the same error message!