Short answer: the grammar is incorrect, but it's such a common mistake that only a grammar nerd or an English learner would notice.
In the structure [ "suggest that" + clause ], the clause has different grammar depending on the meaning of "suggest" in that context.
The meaning in this context is to propose as desirable or fitting. With this meaning, the verb in the clause should be in the subjunctive mood. This means the correct version of this sentence is:
He suggested (that) Anna use these phrases.
This is subjunctive grammar (not an implied or elided "should").
Another meaning of "suggest" is to mention or imply as a possibility. With that meaning, the verb in the clause follows normal subject-verb agreement.
Let us remind ourselves of the phrases he suggested (that) Anna uses.
The meaning of this sentence is, "He said that it's possible that Anna uses these phrases". This is quite different to the intended meaning of, "He told to Anna that she should use these phrases".
That meaning is odd, so here's a more natural example:
Some detectives are trying to solve a string of bank robberies. They don't understand why they the robber isn't on any of the external security camera footage. One of them suggested that maybe she uses different disguises each time once she's outside.
Native speakers --even educated English language professionals-- often make mistakes with the subjunctive, and few people notice. I am a veteran English teacher with a degree in Linguistics, and I had to read that sentence a few times before I noticed anything was wrong, and wouldn't have noticed a problem if you hadn't caught it and pointed it out.