Otdfbt

I have a polygon and a point feature class in some projection. The points are contained within the polygons

The points are always inside the polygons no matter the projection destination I'm projecting points and polygons to (including 'unprojecting' to geographic).

Putting visually JGH example I have a big polygon in WKID =4326 with a point outside of it...

...when I project them (say in 3347 to follow JGH example) I get this...

where the point now is inside the polygon. I used the ArcGIS Pro 'project' function.

When I use project setting the optional parameter 'Preserve Shape' as in the following picture.

...I get something definitely better like...

where (hopefully) the point stays outside the polygon.
'Preserve shape' parameter did what outlined by J.R. Extra vertices that provide a better discretization of the polygon in the destination projection....otherwise everything is reduced to projecting 4 vertices and drawing straight lines in the destination space to connect them (producing the indesiderable output of the second picture above)

Am I getting this right ?

Some software will use great circle arcs to connect unprojected vertices (sometimes when using a special data type, like PostGIS geography), while projected (or unprojected using the geometry datatype) vertices are connected using straight lines.

This can result in a point being inside a polygon expressed as geography but outside of it if expressed as geometry

The following example uses PostGIS. The polygon goes up to latitude 50, the point is at latitude 51.

WITH poly AS (select ST_GeomFromText('polygon((0 0, 50 0, 50 50, 0 50, 0 0))',4326) as geom),
 pnt AS (select ST_GeomFromText('point(25 51)',4326) as geom)
SELECT ST_INTERSECTS(poly.geom,pnt.geom) intersect_geometry,
 ST_INTERSECTS(poly.geom::geography,pnt.geom::geography) intersect_geography
FROM poly, pnt;

 intersect_geometry | intersect_geography
--------------------+---------------------
 f | t

Edit

Extending on @JR comment, here is an example when projecting a huge polygon to a Lambert Conformal Conic projection (3347) and checking the point intersection near the central meridian: we can see a difference of 4 degrees of latitude between the two!

WITH poly AS (select ST_GeomFromText('polygon((-70 45, -130 45, -130 50, -70 50, -70 45))',4326) as geom),
 pnt AS (select ST_GeomFromText('point(-100 54)',4326) as geom)
SELECT ST_INTERSECTS(poly.geom,pnt.geom) intersect_geometry,
 ST_INTERSECTS(st_transform(poly.geom,3347),st_transform(pnt.geom,3347)) intersect_reproject
FROM poly, pnt;

 intersect_geometry | intersect_reproject
--------------------+---------------------
 f | t

Yes. projections will never reproject a point that was inside a polygon in one projection to be outside it in another--unless there's some sort of precision error. I'm not sure what this property is called in geography, but I just realized it's essentially relativistic invariance, which basically says that as time dilates for us, and our coordinate systems are compressed or stretched, no observer in any given frame will disagree on causal ordering of events. Likewise, no matter in what projection an "observer lives", no one will disagree on what points are in what polygons, even if they disagree on exactly how far the points are from the edges of the polygons, and stuff like that.