What kind of word is "place" in "take place"?

Solution 1:

Take place is, in the words of the ‘Longman Student Grammar of Spoken and Written English’, a ‘multi-word verb construction’ consisting of a verb (take) and a noun phrase (place). Such verbs, formed mainly with take, make, have and do, ‘can combine with noun phrases to form idiomatic verbal expressions. In many cases, the combination also includes a following preposition.’ Other multi-word verb constructions that use a verb and a noun phrase are make fun (of), have a look (at) and take care (of).

These are not phrasal verbs. A phrasal verb consists of a verb followed by an adverbial particle such as about, along, out, up, down, in, off, out or up.

Solution 2:

I'd use the term multi-word verb for 'take place' = 'occur' (if nouns such as 'particle board' can be multi-word, why not verbs?) without feeling the need for Longman's 'construction'.

The greater problem - when should these 'verb + direct object'-like strings be analysed as 'verb + DO' and when not - is large and complex. Passivisation tests (see Janus Bahs Jacquet above) and testing with substitution of it for the apparent DO in a coordinated sentence seem to give inconsistent diagnoses:

take the pledge ...John has taken the pledge ... *The pledge has been taken by John ...(?)John took the pledge and Jim took it too

turn the corner ... We have turned the corner ... The corner has been turned ... *We have turned the corner and France has turned it too

turn turtle ... the boat turned turtle ... *turtle was turned ... *the small boat turned turtle and the larger one almost turned it too

There are many of these verbo-nominal constructions which don't both passivise and accept 'echoed it' substitution. Break camp / cover / jail / ranks / wind; catch fire; change colour; cry wolf; do a bunk; fall prey to ... - I've a list containing hundreds.