To me, perusing a few papers , namely those of
Trigg,Charles W., What is recreational mathematics?, doi:10.2307/2689642i ,Mathematics Magazine, Vol. 51, No. 1 (Jan. 1978), pp. 18-21
Bàrtlovà, Tereza , History and current state of
recreational mathematics and its
relation to serious mathematics, Doctoral Thesis, Department of Mathematical Analysis, Charles University in Prague
Rowlett, Peter, Smith, Edward, Corner, Alexander, O'Sullivan, David and Waldock, Jeff,The potential of recreational mathematics to support the
development of mathematical learning, International Journal of Mathematical
Education in Science and Technology, 50 (7), 972-986.
Provide a few pointers on how recreational mathematics should be identified. A question in recreational mathematics :
can have a "spirit of play or fun" about it. This is a fairly broad, but important characterization nevertheless. Examples include mathematical games, puzzles, paradoxes etc.
can be accessible to a large audience so that it is appreciated at the amateur or semi-professional level. Examples include the Königsberg bridge problem solved by Euler, the Archimedes cattle problem etc.
can be linked to "positive feeling and motivation", to the extent that professors can suggest it as alternate questions in a course they offer, completely different from the syllabus but one that keeps students (and adults, especially retired folk) alert and active. For example, to practice proof techniques like induction, students could be asked questions from very different recreational settings (like the famous L-domino tiling problem, or a finite version of Hilbert's hotel) so that they become more versatile in using it.
can be introduced for pedagogical reasons, as an invitation to explore more complex phenomena. Examples include Lego building block models for motivating combinatorial quantities.
can ask for proofs of mathematical results that are small enough in complexity to be accessible to laymen. Typical examples can be found as "explain how ... may have been found without calculators". I have answered a question about how various Mersenne primes were found without the usage of calculator, prior to the invention of digital devices. As I presented theorems and results from the subject in this answer, I think such a question is on-topic. Another example can ask for how Ramanujan discovered a particular complicated identity, and if anything in his notebook suggested this identity.
can come from actual sources of recreational mathematics. Such are in the books of Schaaf, A bibliography of recreational mathematics, for example. There used to be a Journal for recreational mathematics, it lasted a long time before closing in $2014$. There used to be a Recreational mathematics magazine, and some more sources exist, see Trigg's paper and this article as well. Dudley's book and Sam Loyd's book of puzzles are good sources of recreational math as well.
I think the above points substantially cover what the problem statement, source, background of OP, and motivation should look like for a question in recreational math.
In any case, let me summarize the above and write it again. Before that, an obvious caveat.
Context is key, without it you may not be able to distinguish between a question in recreational math and one not in recreational math. In the absence of context, please vote to close.
To expand a little more, while a problem statement involving digits-of-numbers, winning strategies in children games etc. may seem to be recreational mathematics, it is not a priori clear without context that there is an underlying concept that is being tested or honed via the question, which if present certainly provides "mathematical purpose" to the question, rendering it non-recreative.
As I mentioned, there are official sources of recreational mathematics available. This aids classification. In addition, recreational mathematics can have pedagogical or application-based origins, such as modelling a game of football say. What doesn't count as a source of recreational mathematics is a clarification or question from a fairly rigorous mathematical text such as an examination of a mathematical course, an undergraduate mathematics text or a research paper.
Background of poster
Posters can easily have non-existent or minimal background in mathematics. They may not even be interested in the minor details of the solution, but merely whether certain possibilities are true (such as an optimal ratio for maximizing a given formula for the number of satisfied guests at a wedding, or a particular shape having given characteristics such as every pair of adjacent angles adding up to a perfect square, say). On the other hand, professional mathematicians are unlikely to be asking recreational questions given that they wish to be contributing to ongoing research as a motivation for asking their questions (though this is not fully true, but it's good enough).
Naturally, the motivation for questions from a recreational background are mostly "fun" : they are either built upon quirky observations (such as noticing that $a!b!c! = a!+b!+c!$ has a solution and then asking for all of them, say) or come from real-life non-mathematical fun activities such as snakes-and-ladders probabilities, say. The motivation for a textbook or exam question, on the other hand, is a test of mathematical ability which disqualifies such a question from being recreational.
Also note that historical motivations could occasionally be off-topic, but like I remarked earlier there are possibilities of asking good "historical" questions motivated by nothing but a wish for laymen to observe what great mathematicians did without the necessary tools back in those times.
Like I mentioned earlier, recreational mathematical questions are likely to have attempts that are programming-based numerical checks , finding solutions of equations which involve small numbers by hand, and the likes. There is often no attempt to mathematically formulate or attack the problem, but there is evidence of patterns emerging which answerers can take advantage of. Also, the background of the poster can restrict the toolset available to potential answerers (such as not allowing algebraic number theory to be used to solve $x^2+8=y^3$, say), but occasionally posters will also allow the use of pretty much any tool that helps break down the problem better.
On the other hand, in questions which are not recreational, the background will typically indicate that a particular "syllabus" or pattern of rigorous study was in effect prior to the student being acquainted with the question. For the best benefit of future visitors, the optimal answer must adhere to that syllabus and those tools and train the OP and future visitors to recognize and use that specific tool for this situation (such as a model exercise for the Hahn-Banach theorem).
This is my two cents on this issue. Of course the line is very thin, but to me context is the key distinguishing factor between what's recreational and what's not, and I think this write-up makes the key differences clearer.