Jelly, 24 bytes

,ZµŒgaS$€F)Zṣ€0Ṣ<JƊ€€)FẸ 

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A monadic link taking as its argument the grid as a matrix with 1 representing squares needing a number and 0 representing the spacers. Returns 0 if the grid is Latin and 1 if not.

Explanation

Main link, takes a matrix of 1s and 0s as its argument, z , | Pair with: Z | - z transposed µ | Start a new monadic chain ) | For each matrix (z and transpose(z)): ) | - For each row: Œg | - Group runs of equal elements $€ | - For each group: a | - And with: S | - Sum of group F | - Flatten Z | - Transpose ṣ€0 | - Split each at zeros Ɗ€€ | - For each group: Ṣ< | - Check whether sorted group is less than: J | - Sequence along group F | Finally, flatten Ẹ | And check if any group returned true 

Alternative in case true=true/false=false required:

Jelly, 24 bytes

,ZµŒgaS$€F)Zṣ€0Ṣ_JƊ€€)AƑ 

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Prolog (SWI) + CLPFD, 192 bytes

A/B:-transpose(A,B). []*[]*_*1*_. [0|T]*[0|B]*_*1*_:-T*B*L*L*[]. [1|T]*[A|B]*L*C*Z:-all_distinct([A|Z]),A#>0,A#<L,R#=C-1,T*B*L*R*[A|Z]. []-[]. [X|T]-[Y|B]:-X*Y*L*L*[],T-B. +A:-A-B,A/T,B/U,T-U. 

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The main predicate is +/1.

Verbose

t(A,B):-transpose(A,B). h([],[],_,1,_). h([0|T],[0|B],_,1,_):-h(T,B,L,L,[]). h([1|T],[A|B],L,C,Z):-all_distinct([A|Z]),A#>0,A#<L,R#=C-1,h(T,B,L,R,[A|Z]). q([],[]). q([X|T],[Y|B]):-h(X,Y,L,L,[]),q(T,B). g(A):-q(A,B),t(A,T),t(B,U),q(T,U). 

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g/1 takes the grid as a list of lists, each element being a 1 for a cell or a 0 for a hole. q/2 takes a grid as the first argument and unifies the second argument with a solution of the grid. h(A,B,L,L,[]) takes a row of this grid as A, and unifies a solution for the row to B, and q/2 recursively calls h/5 on each of its rows.

h(A,B,L,C,Z) has row A of the grid, a solution B, L is the length of the current length of 1s plus one so that the cells of B can be constrained to the exclusive range (1, L), C is the count of the remaining length so that L is unified to the length, and Z is the solution for the current contiguous sequence of cells so that all_distinct/2 can be called. The end cases are stated as conditions so that these constraints are fulfilled.

Because this is Prolog, converting the main predicate to take two arguments, like A+B or g(A,B), Prolog will unify the solutions with the variable B, thus also allowing you to see all available solutions. It can also generate valid grids, but running this makes Prolog list all grids with empty rows.

R, 169 bytes

function(x)any(unlist(lapply(1:2,function(m)apply(apply(x,m,function(y,r=rle(y))rep(r$l*r$v,r$l)),1,function(a)tapply(a,cumsum(!a),function(d)sort(e<-d[!!d])<seq(e)))))) 

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An R translation of my Jelly answer. I’m not sure it’s much easier to follow! Takes a matrix of 0s and 1s and returns a logical scalar (FALSE if Latin, TRUE if not). Note because of the way base R tends to collapse matrices to vectors, the input has to have at least 2 rows and columns, but this can be done by padding input with an extra row or column of FALSEs. If this is unacceptable, I can work around it at the cost of extra bytes to handle the edge case.