Learn English – Difference between “matching” and “overlapping interests”

It is possible to say that interests overlap, but I wouldn't interpret them the same way.

John and I have matching interests.

If you said this, I would interpret it to mean that you and John share most, if not all, of the same interests. Your interests match – they are the same. See this definition (from Oxford Dictionaries):

match
Verb

1 Correspond or cause to correspond in some essential respect; make or be harmonious:
[with object] "I thought we'd have primrose walls to match the bath"
[no object] "the jacket and trousers do not match"

John and I have overlapping interests.

I would interpret this to mean that you and John share some interests, but not all and probably not even a significant portion of interests. This is consistent with the dictionary definition:

overlap
Verb

1 Extend over so as to cover partly:
"the canopy overlaps the house roof at one end"
[no object] "the curtains overlap at the centre when closed"

1.1 [no object] Cover part of the same area of interest, responsiblity, etc.:
"the union's committments overlapped with those of NATO"

Your examples sentences, I think, would be better if the order were reversed (and rephrased to use the simple present rather than continuous):

The research of Mr. X matches my interests.

The research of Mr. X overlaps with my interests.

You could also say "Mr. X's research..." but that would be down to writing style.

When to match is used in the present continuous (is/are matching), it sounds like the following definition is being used instead of your intended definition:

match
Verb

1.1 [with object] Put (someone or something) together with someone or something else appropriate or harmonious:
"she was trying to match the draperies to the couch"

Use of the continuous of to overlap just doesn't sound right in this context, although I can't provide further explanation as to why.