I know that in the following phrase I must add the letter s to the verb seem when used with third person singular like this:
It seems to me...
But then I see such phrases as
make it seem...
and I'm entirely baffled why the word seem is used without the letter s. Isn't seem used with third person singular here?
Note: English is my third language, but I have a good command of it. Still such peculiarities are perplexing.
In your example, the verb form of "seem" in "make it seem" is the so-called "bare infinitive" which is required when you use the word "make" with the meaning "to cause someone to do something".
You can read about this use of "make" here (and lots of other places too).
One of the examples given there is "His mother made him clean his room."
Similarly, the phrase "make it seem" would be part of a longer sentence, maybe something like: "The yellowed paper and the mildew stains make it seem like an ancient document, but it's a forgery." In that example, "The yellowed paper and the mildew stains" is the subject of the sentence. "Make" agrees with that subject, and "seem" is a bare infinitive as discussed in the reference above.
-s: "The yellowed paper makes it seem like ..."
When using the third person, you would add the s to the word make, not seem.
For example:
You make it seem easy, I make it seem easy. She makes it seem easy, too.