manufacturer も maker も「[メーカー](https://eikaiwa.dmm.com/uknow/questions/35013/)」ですが、前者の方がフォーマルです。
会社名は言わなくても、I work for a medical equipment maker. (私は医療機器のメーカーで[働いています](https://eikaiwa.dmm.com/uknow/questions/75261/)。）と言うこともできます。
You never have to give the details of your company, but you could say "I work in manufacturing" or "I work in a manufacturing company." If you want to give more information about the type of manufacturing you could say "I work in a manufacturing company that specializes in (automotive parts, etc.)".
会社の詳細を言う必要はありませんが、「I work in manufacturing（製造業界で働いています」 や「I work in a manufacturing company.（製造業の会社で働いています）」ということができます。
製造業のより多くの情報を出したい場合は、「"I work in a manufacturing company that specializes in (automotive parts, etc.).」製造業の会社で働いています特に（自動車の部品など）です。
If you describe your employer as 'a well known manufacturer' it is quite clear (unless the listener is an idiot!) that you do not wish to mention your employer's name.
To avoid any reference to your employer, it may be less troublesome to simply say, "I work in the manufacturing field" or "I work in the manufacturing industry,"
'a well known manufacturer'と自分の雇い主を表現したら、相手が無知でなければ名前までは言いたくないんだというのを相手もわかると思います。
どうしても自分の会社をいいたくなければ、"I work in the manufacturing field" または "I work in the manufacturing industry,"と簡単にいえば面倒を避けられると思います。
*I work for a manufacturing company.
This is a good way of expressing yourself without providing the company name.
For example:
Stranger: Where do you work?
You: I work for a manufacturing company.
If you see that they keep asking you the details, you can change the topic.
*I work for a manufacturing company.
これはあなたの会社名を言わずにあなた自身を表現する良い方法です。
例：
Stranger: Where do you work?
ー不審者：あなたはどこで働いているの？
You: I work for a manufacturing company.
ーあなた：私は製造会社で働いているよ。
もし彼らが詳細について詳しく聞きたがるようであれば、トピックを変えましょう。
In the first sentence, you have said that you work for a manufacturing company, but you have not mentioned the name of the company or the name of the product that the company manufactures. Perhaps, mentioning the name of the product would have given away the name of the company because sometimes, the product's name can be included in the name of the company.
In the second sentence, you have said that your company is in the manufacturing industry, again, you have not mentioned the name of the company.
You could use word like
" I work for a craftsman".
This usually means a person that makes hand made products.
"I work for a Producer"
Somebody who produces things
"I work for a Constructor".
I work in the fabrication field or the building field or a word used to describe the
industry you work in,
Building Industry
Construction
Computing etc... without giving the company name.
次の表現も使うことが出来ます。
【例】
" I work for a craftsman"
（職人として働いています）.
これは普通手作りの商品をつくっている人の事を言います
"I work for a Producer"
（生産者として働いています）
何かを生産する人です。
"I work for a Constructor"
（建設家として働いています）
"I work in○○"（～分野で働いています）
"the fabrication field"（製造分野）
the building field （建築分野）
これらの言葉は産業などを言い表しています。
"industry you work in"（あなたが働く産業）
Building Industry（建築産業）
Construction Industry (建築産業）
Computing Industry （コンピューター産業）
など
会社名を言うことなく言い表すことが出来ます。
You work for a company that manufactures things, but you do not want to say the name of the company, then you can use this sentence to explain.
For example
How is everything?
Everything is good.
Where do you work again?
I work for a manufacturing company.
What is it called?
That isn't important. Hungry?
I hope this helps.
Have a great day.
Will
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