Here in 2022, you can markedly simplify that code using async/await, like this:

private async bindSemanticObject(unom: number) {
    try {
        const addressSemantic = await this.setSemanticService
                .bindObjectsSemanticByUnom(unom);
        this.setSemanticService.setAddressSemantic(addressSemantic);
        this.setSemanticService.verifySemanticObject();
        this.setSemanticService.setSemanticFields();
        await this.setSemanticService.saveSemantic();
        try {
            await this.setSemanticService
                .stateChangeBindObject()
                .toPromise();
        } catch {
            // You've said in a comment you don't care about
            // rejections on the part above
        }

        this.toastrService.success('Updated...');
        this.editLayerFactory.destroy(false);
    } catch (e) {
        this.toastrService.warning(e);
    }
}

A few notes on that:

• You've said that you don't care about rejections from this.setSemanticService().stateChangeBindObject.toPromise(), so I've wrapped it in an inner try/catch.
• You've also said that all (other) errors should be shown via toastr, even though the code in the question doesn't do that.
• The outer try/catch will catch both synchronous errors thrown by the code and promise rejections (because of await).

If you can't use async/await for whatever reason, you can use a series of then handlers:

private bindSemanticObject(unom: number) {
    return this.setSemanticService
            .bindObjectsSemanticByUnom(unom)
            .then(addressSemantic => {
                this.setSemanticService.setAddressSemantic(addressSemantic);
                this.setSemanticService.verifySemanticObject();
                this.setSemanticService.setSemanticFields();
                return this.setSemanticService.saveSemantic();
            })
            .then(() => {
                return this.setSemanticService
                    .stateChangeBindObject()
                    .toPromise()
                    .catch(() => { /*...suppress...*/ });
            })
            .then(() => {
                this.toastrService.success('Updated...');
                this.editLayerFactory.destroy(false);
            })
            .catch(e => {
                this.toastrService.warning(e);
            });
}

That's not quite the same, it doesn't handle a synchronous error from the initial this.setSemanticService.bindObjectsSemanticByUnom(unom). If you need to handle that, change the beginning to:

private bindSemanticObject(unom: number) {
    return Promise.resolve().then(() =>
        this.setSemanticService
            .bindObjectsSemanticByUnom(unom)
    )
    .then(addressSemantic => {
        // ...