French has typographic rules and orthographic rules but when talking about pronunciation, you'd rather talk about usages, not rules.
These usages vary depending on several factors and make what is called an accent. Accents are acquired at very early stages of language learning. Kids have usually the same accent as the people surrounding them (family, schoolmates) and then their accent evolves with time, being more or less affected by the surrounding accents. This is not specific to French, all languages spoken in a large enough territory present the same kind of geographical and chronological variations.
That said, some phonemes are identical or almost identical whoever native French people use them while other phonemes present variants. Many vowels are in the second category.
A second thing you need to know about French is the fact it is not written phonetically. The way a word is written only gives hints about how it is pronounced, actual usage might be different to what would be expected. English is worse than French on that subject.
The two letters "ai" might be pronounced /ε/, /e/, /ɛː/, /ɛɪ̯/ (Québec) or even /ə/ in a few words (e.g. faisan).
In aime, ai is almost always pronounced /ε/ in France.
In aimer, the /e/ pronunciation is the most common in Southern France but competes with /ε/ elsewhere.
This ai is often pronounced the same way as the et of poulet. Here is a map from Mathieu Avanzi showing this kind of regional variations, it might not exactly match the aimer figures but is nevertheless an example of such patterns :
Source: https://francaisdenosregions.com/2017/07/06/ces-mots-qui-ne-se-prononcent-pas-de-la-meme-facon-dun-bout-a-lautre-de-la-france/
As you are not a native French speaker, you shouldn't really care about which of the common pronunciations to choose when pronouncing aimer.
The primary goal of oral communication is to be understood, either of these vowels is fine.
