Preface: I'm not sure of the extent to which the commonly used English terms for this service might vary by country. I can only share the US perspective and I'll leave it to other readers to respond regarding any differences in other places.
There are two aspects to this--the "technical" term for someone who performs the service, and the term for the service, itself. I'll try to address both.
Some brief context: The kind of minor repairs and alterations described in the question tend not to be a lucrative business, so there aren't a lot of shops specializing in it. People who go to a shop to obtain the service generally get it at a business specializing in something else and the service is offered as a supplement.
Minor repairs and alterations are frequently services offered by dry cleaning establishments. The work is typically performed by someone who is skilled but not a formally trained professional. So the commonly used terms in that case often refer to the service rather than the person doing it.
The sign on the store and the terms used in advertisements will refer to it as alterations or repairs or a combination of the two, like permutations of "alterations and repairs" sometimes in combination with other related words, like "clothing". If the sign only says "alterations" the public would assume that includes repairs, and vice versa.
There are some shops that specialize in just such minor work, and the signage and advertising would generally be similar.
At the other end of the spectrum, you could get clothing altered or repaired at a shop that specializes in more complex and comprehensive work. In that case, the work is generally performed by someone with formal training, or an apprentice under their supervision, or sometimes a person without formal training who has become very skilled. The terminology used generally refers to the person doing the work, including on signage and in advertising.
A tailor would specialize in making clothing or major alterations to clothing, but would typically also offer minor repairs and alterations. The term is often associated with men's clothing, but it can apply to clothing of either gender. Even if a tailor or tailor shop specializes in men's clothing, they are likely to offer, or at least agree to perform if requested, repairs and alterations on women's clothing, as well.
Dressmakers specialize in making women's clothing, and major alterations to them, and would usually also offer minor repairs and alterations. They will often offer, or agree to perform if requested, minor alterations and repairs on men's clothing, as well.
So if you are just looking for the most generic term for a person or establishment that performs minor repairs and alterations on any clothing, "tailor" is probably your best bet. For terms that would be on the signage or in the advertising of a shop that offers these services, it would likely be "alterations", "repairs", "tailor", or "dressmaker".
 A person whose occupation is making or altering garments (such as suits, jackets, and dresses) typically to fit a particular person. (M-W) Some dictionaries add a qualifier such as "especially men's clothing".
 A person whose occupation is the making or alteration of women's dresses, coats, etc.