Meaning of "Teaming"

Solution 1:

When you see "teaming with", it is fairly often a misspelling of "teeming with", rather than a use of the verb "team (something) with (something)" that you mention in your question.

The sentence you quote contains an example of this misspelling. With correct spelling, the sentence would be "It was the school holidays, and the place was teeming with people, families, youth groups, holiday clubs." "The place was teeming with people" means "The place was full of people". Collins has an explanation in its entry for the verb teem:

verb [usually cont]

If you say that a place is teeming with people or animals, you mean that it is crowded and the people and animals are moving around a lot.

For most of the year, the area is teeming with tourists. [VERB + with]

The words "team" and "teem" are homophonous, and apparently etymologically related, but my understanding is that the spelling "teaming" is not considered standard in this context.

See Merriam-Webster's "Usage Notes: 'Teeming' vs. 'Teaming'":

Each of these words has a number of meanings, spread across multiple parts of speech. The meanings which are most often mixed are “to become filled to overflowing” (teem) and “join forces or efforts” (team), especially when found as teeming and teaming. When one of these words is employed incorrectly it most often seems to be teaming used where teeming is called for.

My thoughts then drift to warmer days of spring and summer with green grass between our toes, birds chirping and calling in the fields around us, lakes and streams teaming with fish and the sights and sounds of a good thunderstorm dropping much-needed rain. — Dan Ackerman, Bismarck Tribune (Bismarck, ND), 18 Jan. 2018

St. Vincent is a lush, volcanic island with magnificent rain forests and waterfalls, while the Grenadines are coral islands with pristine beaches and reefs teaming with life. — The Globe and Mail (Toronto, Can.), 7 Sept. 1996

The Google Ngram Viewer shows that "teeming with" is far more frequent in writing than "teaming with" even in recent years:

Some examples where "teaming with" is the correct spelling

Here are some examples from Google Books of situations where the spelling "teaming with" would be correct:

• Teaming with Your Therapy Dog (book title)

• "Teaming with local teams is considerably easier than with remote teams." (A Commonsense Approach to Dealing with People: Managing People Made Easier, by Terry Jenkins, p. 32)

• "A project would only support client innovation if it included teaming with clients." (Teaming: How Organizations Learn, Innovate, and Compete in the Knowledge Economy, by Amy C. Edmondson)

• "Initially believed to be an office stooge by Hayes and Gordy, Roberts fit right in — he could drink and party as well as any kid he was teaming with and his nonchalant attitude towards everything became his trademark." (The Pro Wrestling Hall of Fame: The Tag Teams, by Greg Oliver, p. 49)

cobaltduck left a comment saying that "the more common way of expressing "teaming with" where the with-an-a spelling is correct, is "teaming UP with". This seems to be supported by another Google Ngram Viewer chart:

You can see that "teaming up with" seems to be more frequent than "teaming with", and because the line for "teaming with" presumably includes many misspellings that should be written as "teeming with", the real difference in frequency between (correctly spelled) "teaming with" and "teaming up with" is presumably even greater. We seem to see a difference in the same direction if we look at some examples with pronouns (although the absolute frequencies are rather low, so the charts look like they have a lot of noise):