Since Chrome has a meaning in the hypothetical, first I would check the trademark registration file for a disclaimer of "Chrome."
Although it's impossible to say with the very limited hypothetical facts, I'd be surprised to have a mark issue for a fanciful word when that word is already registered as part of another.
One big fact that is missing is the similarity of the markets/products between the two registrants. Google brand dessert topping would have a much better chance of being registered than would Google brand online backup services.
Yes, the 1st word is considered more important than the 2nd word, which is considered more important than the 3rd word, etc.,, but a trademark registration protects each word that's not disclaimed by the applicant.
In your hypothetical, if the goods or services were the same or similar or related by the 2 Google users, no one could register "Google" if "Google" were already the 1st word of another TM -- consumers would assume that Google Chrome was a a version of Google.
I'm not sure what you mean when you state that these words do or don't have "meaning." "Secondary meaning," where consumers have developed an association between the TM and the goods/services? Descriptive, where the word describes the goods or services?
You need to see your own TM lawyer for specific help.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.