I realize that the different behavior of equality (__eq__) can be actually a major concern, why such code snippet is probably not good.

However, you could maybe still do this:

if sys.version_info[:2] >= (3, 6):
  OrderedDict = dict
else:
  from collections import OrderedDict as _OrderedDict

  class OrderedDict(_OrderedDict):
    __eq__ = dict.__eq__
    __ne__ = dict.__ne__

The difference is that with the original collections.OrderedDict, {1:1,2:2} is not the same as {2:2,1:1}, but for dict and my overwritten OrderedDict in this example, it is the same.

Some behaviour remains the same, but OrderedDict are reversible and dict are not:

from collections import OrderedDict

d = { "a" : 1, "c" : 2}

od = OrderedDict(d.items())

print(list(reversed(od)))
print(list(reversed(d)))

Output

['c', 'a']

Traceback (most recent call last):
  File "path/to/file", line 8, in <module>
    print(list(reversed(d)))
TypeError: 'dict' object is not reversible

From the documentation:

In addition to the usual mapping methods, ordered dictionaries also support reverse iteration using reversed().

Both OrderedDict and dict are insertion-ordered¹ for iteration. There is practically no reason to use OrderedDict if iteration order is the only deciding point, especially if re-ordering is not needed.
Obviously, if comparison order is desired OrderedDict and dict are not interchangeable.

Or phrased differently, for Python >=3.6, is there any reason to use OrderedDict?

These days OrderedDict is to dict what deque is to list, basically. OrderedDict/deque are based on linked lists² whereas dict/list are based on arrays. The former have better pop/move/FIFO semantics, since items can be removed from the start/middle without moving other items.

Since arrays are generally very cache friendly, the linked list advantage only comes into play for very large containers. Also, OrderedDict (unlike deque) does not have guarantees for its linked list semantics and its advantage may thus not be portable. OrderedDict should primarily be used if many pop/move/FIFO operations are needed and benchmarking can compare the performance of dict vs. OrderedDict in practice.

¹This applies to all currently supported implementations compliant with the Python language spec, i.e. CPython and PyPy since Python 3.6.

²OrderedDict in CPython still preserves O(1) key access. This is realised by also having a "regular" lookup table, using the linked list for order between items and the lookup table for direct item access. It's complicated.