We need to see stems and buds close up.

geographical location? Big Box Store, Lowes or Home Depot perhaps. Looks like a cultivar of Euonymus japonica.

Head back the current season's growth in summer. If you really need it to be flat against the house - maybe that was the idea with the original planting - and there are enough branches back there already cut the branches in front of this back layer off during winter. And then going forward from that point prune each summer to maintain the new or restored shape, cutting off attempts to regain the previous structure at the ground. And shortening shoots coming from the front of the retained section that point away from the wall. If instead the base of the shrub is located well out away from the wall then maybe you'll just have to remove it. The fundamental issue being the shrub is too large growing a kind for that planting site.

Really how close are the neighbours? IMO a fence creates enough privacy for most people and any kind of tree has roots and with all that hardscape those roots will never get enough water so they will be looking for water where ever they can find it which could be right under the hardscape.IMO too much paver in relation to green space .