I'm bad at remembering songs names and lyrics.

During my childhood, I often listened to English songs all the time since every member of my family have thisd/has a taste for international music.

"Often" and "all the time" have similar meanings, so you don't need both here.

I'm not sure that not knowing the language made me loste interest in the lyrics, or if I was just enjoying the melody and didn't care about them anyways.

When someone asks me what genre of music I like the most, I can't give it a simple answer.

Sometimes I feel embarrassed when everyone is singing along to a very popular song but I can't join them because I never memorized the lyrics.

There are exceptions, like when a song has a deep connection within my life or when I stop to literalfully/thoroughly/completely study the lyrics until I memorize them, which takes me days to sink in.

I think the word you're really looking for here is something that describes you trying to learn song lyrics in a lot of detail.

I'm bad at remembering songs nam titles and lyrics.

Don't need the s on songs at the end. song describes names and lyrics.

Generally, I pay more attention to the melody of songs instead ofrather than the lyrics, because the mood and the vibe of the of music are more important to me than the lyrics.

During childhood, I often listened to English songs all the time since every member of my family have thisa taste for international music.

"taste" is a generic thing so indefinite article.

The problem was that I didn't know English sufficiently to understand the lyrics by myself,: just some words here and there.

Second part gives further details so :

I'm not sure thatif not knowing the language made me loste interest in the lyrics, or if I was just enjoyinged the melody and didn't care about them anyways.

"if" Conditional if... or if..... It is "lose interest" not "lost interest". "enjoyed" same tense as "didn't care" i.e. simple past. "anyways" is very slangy sounding "anyway" is better.

When someone asks me what genre of music I like the most, I can't give it a simple answer.

Don't need the "it there is no-one or no thing that you are talking to.

I usually reply with "Mmostly 80's music" to make it easier.

In fact, I like 80's music, but I always get anxious because there's a chance the person knows somthe names of some famous bands and I cawon't be able to keep going with the conversation.

Sometimes I feel embarrassed when everyone is singing along ato very popular song but I can't join themin because I never memorized the lyrics.

"to sing along to". You can replace "lyrics" with "words" if you wish it would probably be more common.

There are exceptions, like when a song has a deep connection withto my life or when I stop to literally study the lyrics until I memorize them, which takes me days to sink in.

"which takes me days" or "which takes days to sink in".