No. Your passport does not need to have at least 6 months of remaining validity to apply for a UK ETA.
Under UK entry requirements, your passport must be valid for the entire duration of your stay in the United Kingdom.
Therefore, even if your passport has less than 6 months of validity remaining, you can apply for a UK ETA and visit the UK as long as your passport remains valid throughout your planned stay.
For example, if your passport expires in 4 months and you plan to visit the UK for 10 days, you can travel with a UK ETA, provided you meet all other entry requirements.
However, a UK ETA is valid for up to 2 years from the date of approval or until your passport expires, whichever comes first.