Your https://geo.api.vlaanderen.be/VKBO/wfs?service=WFS&request=GetCapabilities&typename=VKBO:Vkbo&PROPERTY_NAME=”VKBO_NISCODE”&CQL_FILTER=VKBO_NISCODE=%2744048%27
works in giving a GetCapabilities response for the highest version of service supported (2.0.0) because you didn't specify version or acceptversions parameters.
The GetCapabilities request may accept a sections parameter, but is not expecting typename or PROPERTY_NAME or CQL_FILTER and ignores them.
https://geo.api.vlaanderen.be/VKBO/wfs?request=GetFeature&service=WFS&version=1.1.0&typename=VKBO:Vkbo&PropertyName=VKBO_NISCODE
gives results as expected; all values for just PropertyName VKBO_NISCODE for typename VKBO:Vkbo.
If CQL_FILTER used to work but now doesn't it's possible that the old server supported CQL filters and the new server doesn't. CQL filtering may be offered as an extra capability for a WFS, it isn't part of the WFS + Filter Encoding standard.
A FE filter to get your property value would be:
Filter=<Filter><PropertyIsEqualTo><PropertyName>VKBO_NISCODE</PropertyName><Literal>44048</Literal></PropertyIsEqualTo></Filter>
You need to URL encode that in GetFeature GET request like:
https://geo.api.vlaanderen.be/VKBO/wfs?request=GetFeature&service=WFS&version=1.1.0&typename=VKBO:Vkbo&PropertyName=VKBO_NISCODE&Filter=%3CFilter%3E%3CPropertyIsEqualTo%3E%3CPropertyName%3EVKBO_NISCODE%3C%2FPropertyName%3E%3CLiteral%3E44048%3C%2FLiteral%3E%3C%2FPropertyIsEqualTo%3E%3C%2FFilter%3E
or perhaps you want all values of the typename VKBO:Vkbo when the VKBO_NISCODE PropertyName is 44048 like:
https://geo.api.vlaanderen.be/VKBO/wfs?request=GetFeature&service=WFS&version=1.1.0&typename=VKBO:Vkbo&Filter=%3CFilter%3E%3CPropertyIsEqualTo%3E%3CPropertyName%3EVKBO_NISCODE%3C%2FPropertyName%3E%3CLiteral%3E44048%3C%2FLiteral%3E%3C%2FPropertyIsEqualTo%3E%3C%2FFilter%3E
https://geo.api.vlaanderen.be/VKBO/wfs?request=GetFeature&service=WFS&version=1.1.0&typename=VKBO:Vkbo&PropertyName=VKBO_NISCODE
gives results as expected. If CQL_FILTER used to work but now doesn't it's possible that the old server supported CQL filters and the new server doesn't. CQL filtering may be offered as an extra capability for a WFS, it isn't part of the WFS + Filter Encoding standard.