Is there a word to describe someone who has the same taste/likes as you? For example, I like dogs. How can I describe all the members in a FB group that's dedicated to sharing cute dog images/videos? The closest phrase I can think of is "interest group", but that seems to have a very specific meaning and usage.
The phrase I associate the most with this is like-minded. From the Free Dictionary
agreeing in opinions, goals, etc
It doesn't have the institutional connotations that you are seeking to avoid.
The second definition of sympathy, same source, usually in the plural form for this usage, also rings true:
a. Mutual understanding or feeling between people: "Like two frightened children, we sought at the same time to comfort one another, so quick was the sympathy between us" (Nicholas Meyer).
b. Agreement with or support for an opinion or position: The mayor is in sympathy with the proposal.
c. often sympathies A tendency to support a position or opinion: a politician of conservative sympathies.
Kindred spirit(s): a person(or persons) who has the same opinions, feelings, and interests as you. https://dictionary.cambridge.org/us/dictionary/english/kindred-spirit
I would recommend
common-interest friends ,
who are "friends that a person has and keeps in contact with because of a shared interest, hobby, activity, club, etc." (Link)
I like congenial: "having the same nature, disposition, or tastes".