Learn Russian: У Анны есть своя зарядка.

In Russian, when the person who owns the object is the main focus of the sentence, we use forms of свой (meaning "one's own"). In a "someone has" construction like У Анны есть, Anna is the logical owner. Using своя makes it clear it is her own charger. If you said У Анны есть её зарядка, it would sound like Anna has some other woman's charger.

Why is своя in the nominative case instead of the accusative case?

In the Russian "to have" construction (У ... есть), the thing you have is actually the grammatical subject of the sentence, which means it takes the nominative case. Since зарядка is a feminine noun in the nominative case, the pronoun describing it must be the feminine nominative form, своя.

Why is Anna's name spelled Анны instead of Анна?

The preposition у always requires the genitive case. For feminine nouns ending in -а like Анна, the genitive ending is -ы. So, У Анны есть literally translates to "By Anna there is".

I have seen the word зарядка used for "morning exercise". Does it also mean "charger"?

Yes! Originally, зарядка means "charging" or "recharging" (which is why it is used for morning exercises that give you energy). In modern everyday Russian, it is the most common word for a phone or laptop charger.